I 


AN   INTRODUCTION  TO 
ELECTRODYNAMICS 


FROM  THE  STANDPOINT  OF  THE 
ELECTRON  THEORY 


BY 

LEIGH  PAGE,  PH.D. 

ASSISTANT    PROFESSOR   OF   PHYSICS   IN  YALK«UNIVE;ag1i'l\Y   ' 


GINN  AND  COMPANY 

BOSTON     •     NEW   YORK     •     CHICAGO     •     LONDON 
ATLANTA     •     DALLAS     •     COLUMBUS     •     SAN   FRANCISCO 


COPYRIGHT,  1922,  BY 
LEIGH  PAGE 


ALL   RIGHTS    RESERVED 

822.2 


GINN   AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


PREFACE 

The  object  of  this  book  is  to  present  a  logical  development 
of  electromagnetic  theory  founded  upon  the  principle  of  rela- 
tivit}7'.  So  far  as  the  author  is  aware,  the  universal  procedure 
has  been  to  base  the  electrodynamic  equations  on  the  experi- 
mental conclusions  of  Coulomb,  Ampere,  and  Faraday,  even 
books  on  the  principle  of  relativity  going  no  farther  than  to 
show  that  these  equations  are  covariant  for  the  Lorentz -Einstein 
transformation.  As  the  dependence  of  electromagnetism  on  the 
relativity  principle  is  far  more  intimate  than  is  suggested  by 
this  covariance,  it  has  seemed  more  logical  to  derive  the  electro- 
dynamic  equations  directly  from  this  principle. 

The  analysis  necessary  for  the  development  of  the  theory  has 
been  much  simplified  by  the  use  of  Gibbs'  vector  notation. 
While  it  is  difficult  for  those  familiar  with  the  many  conven- 
iences of  this  notation  to  understand  why  it  has  not  come  into 
universal  use  among  physicists,  the  belief  that  some  readers 
might  not  be  conversant  with  the  symbols  employed  has  led  to 
the  presentation  in  the  Introduction  of  those  elements  of  vector 
analysis  which  are  made  use  of  farther  on  in  the  text. 

Chapter  I  contains  a  brief  account  of  the  principle  of  rela- 
tivity. In  the  second  chapter  the  retarded  equations  of  the 
field  of  a  point  charge  are  derived  from  this  principle,  and  in 
Chapter  III  the  simultaneous  field  of  a  moving  charge  is  dis- 
cussed in  some  detail.  In  the  next  chapter  the  dynamical  equa- 
tion of  the  electron  is  obtained,  and  in  Chapter  V  the  general 
field  equations  are  derived.  Chapter  VI  takes  up  the  radiation 
of  energy  from  electrons,  and  Chapters  VII  and  VIII  contain 
some  applications  of  the  electromagnetic  equations  to  material 
media,  chosen  as  much  for  their  illustration  of  the  theory  as 
for  their  fundamental  importance.  Throughout,  great  pains 


iv          AN  INTRODUCTION  TO  ELECTRODYNAMICS 

have  been  taken  to  distinguish  between  definitions  and  assump- 
tions, and  to  carry  on  the  physical  reasoning  as  rigorously  as 
possible.  It  is  hoped  that  the  book  may  be  found  useful  by 
those  lecturers  and  students  of  electrodynamics  who  are  looking 
for  a  logical  rather  than  a  historical  account  of  the  science. 
The  subject  matter  covers  topics  appropriate  for  a  one-year 
graduate  course  in  electrodynamics  and  electromagnetic  theory 
of  light. 

The  author  wishes  to  acknowledge  his  debt  to  those  great 
thinkers,  Maxwell,  Poynting,  Gibbs,  Lorentz,  Larmor,  and  Ein- 
stein, and  to  express  his  appreciation  of  the  inspiration  and  un- 
failing interest  of  his  former  teacher,  Professor  H.  A.  Bumstead. 
His  thanks  are  due  his  colleague,  Professor  H.  S.  Uhler,  for  many 
suggestions  tending  toward  greater  clearness  of  exposition. 

LEIGH  PAGE 
YALE  UNIVERSITY 


CONTENTS 


INTRODUCTION.     ELEMENTS  OF  VECTOR  ANALYSIS 

PAGE 

ADDITION  AND  MULTIPLICATION 1 

GAUSS'  THEOREM 4 

STOKES'  THEOREM 5 

DYADICS 6 

CHAPTER  I.     THE  PRINCIPLE  OF  RELATIVITY 

MOTION 10 

REFERENCE  SYSTEM 10 

PRINCIPLE  OF  RELATIVITY 11 

RECIPROCAL  SYSTEMS 13 

DIFFERENTIAL  TRANSFORMATIONS 13 

SPACE  AND  TIME  TRANSFORMATIONS 16 

FOUR-DIMENSIONAL  REPRESENTATION 18 

CHAPTER  II.  THE  RETARDED  FIELD  OF  A  POINT  CHARGE 

ELECTRIC  FIELD 20 

MOTION  OF  A  FIELD 21 

TRANSFORMATION  EQUATIONS 21 

POINT  CHARGE  AT  REST 25 

POINT  CHARGE  IN  MOTION 27 

RETARDED  POTENTIALS 30 

CHAPTER  III.     THE  SIMULTANEOUS  FIELD  OF  A  POINT  CHARGE 

CONSTANT  VELOCITY 33 

CONSTANT  ACCELERATION 34 

GENERAL  CASE 38 

CHAPTER  IV.     THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON 

ELECTRICAL  THEORY  OF  MATTER 42 

DYNAMICAL  ASSUMPTION 43 

CONSTANT  VELOCITY 44 

v 


vi         AN  INTRODUCTION  TO  ELECTRODYNAMICS 

PAGE 

CONSTANT  ACCELERATION 46 

GENERAL  CASE 50 

RIGID  BODY 54 

EXPERIMENTAL  DETERMINATION  OF  CHARGE  AND  MASS  OF 

ELECTRON 56 

CHAPTER  V.     EQUATIONS  OF  THE  ELECTROMAGNETIC  FIELD 

DIVERGENCE  EQUATIONS 59 

VECTOR  FIELDS 60 

CURL  EQUATIONS 62 

ELECTRODYNAMIC  EQUATIONS 63 

ENERGY  RELATIONS 64 

ELECTROMAGNETIC  WAVES  IN  SPACE 65 

RADIATION  PRESSURE 67 

ELECTROMAGNETIC  MOMENTUM 73 

FOUR-DIMENSIONAL  REPRESENTATION 74 

CHAPTER  VI.     RADIATION 

RADIATION  FROM  A  SINGLE  ELECTRON  .    '. 79 

RADIATION  FROM  A  GROUP  OF  ELECTRONS 80 

ENERGY  OF  A  MOVING  ELECTRON 81 

DIFFRACTION  OF  X  RAYS 85 

CHAPTER  VII.    ELECTROMAGNETIC  FIELDS  IN  MATERIAL  MEDIA 

FUNDAMENTAL  EQUATIONS 89 

SPECIFIC  INDUCTIVE  CAPACITY 99 

MAGNETIC  PERMEABILITY 101 

ENERGY  RELATIONS 105 

METALLIC  CONDUCTIVITY 106 

REDUCTION  OF  THE  EQUATIONS  TO  ENGINEERING  FORM      .     .     .  108 

CHAPTER  VIII.    ELECTROMAGNETIC  WAVES  IN  MATERIAL  MEDIA 

ISOTROPIC  NON-CONDUCTING  MEDIA 112 

ANISOTROPIC  NON-CONDUCTING  MEDIA  .     .     .  • 115 

REFLECTION  AND  REFRACTION 122 

ROTATION  OF  THE  PLANE  OF  POLARIZATION 127 

METALLIC  REFLECTION 130 

ZEEMAN  EFFECT 132 


AN  INTRODUCTION  TO 
ELECTRODYNAMICS 

INTRODUCTION 

ELEMENTS  OF  VECTOR  ANALYSIS 

Addition  and  multiplication.  A  vector  is  defined  as  a  quantity 
which  has  both  magnitude  and  direction.  It  will  be  designated 
by  a  letter  in  blackface  type,  its  scalar  magnitude  being  repre- 
sented by  the  same  letter  in  italics.  Geometrically,  a  vector  may 
be  represented  by  an  arrow  having  the  direction  of  the  vector 
and  a  length  proportional  to  its  magnitude.  The  beginning  of  this 
representative  straight  line  is  known  as  its  origin,  and  the  end, 
as  its  terminus.  To  add  two  vectors  P  and  Q  place  the  origin 
of  Q  at  the  terminus  of  P.  Then  the  line  drawn  from  the 
origin  of  P  to  the  terminus  of  Q  is  defined  as  the  sum  of  P 
and  Q.  To  subtract  Q  from  P  reverse  the  direction  of  Q  and 
add.  The  components  of  a  vector  are  any  vectors  whose  sum  is 
equal  to  the  original  vector.  Although,  strictly  speaking,  the 
components  of  a  vector  are  themselves  vectors,  the  term  com- 
ponent will  often  be  used  to  denote  the  magnitude  alone  in 
cases  where  the  direction  has  already  been  specified. 

A  vector  is  often  determined  by  its  components  along  three 
mutually  perpendicular  axes  X,  Y,  Z.  These  axes  will  always 
be  taken  so  as  to  constitute  a  right-handed  set ;  that  is,  so  that 
a  right-handed  screw  parallel  to  the  Z  axis  will  advance  along 
this  axis  when  rotated  from  the  X  to  the  Y  axis  through  the 
right  angle  between  them.  Let  i,  j,  k  be  unit  vectors  parallel 

1 


2-         A^^I^TiiODUCTION  TO  ELECTRODYNAMICS 

respectively  to  the  X,  F,  Z  axes.    Then  if  the  projections  of  P 
along  these  axes  are  denoted  by  j£,  Py,  Pz, 


and,  obviously, 

P  +  Q  =  «  +  eji  +  «  +  ^)3  +  «  +  e,)k.  (2) 

If  two  or  more  vectors  are  parallel  to  the  same  straight  line, 
they  are  said  to  be  collinear.  If  three  or  more  vectors  are  par- 
allel to  the  same  plane,  they  are  said  to  be  coplanar. 

Two  vectors  P  and  Q  may  be  multiplied  together  in  three 
different  ways.  The  most  general  type  of  multiplication  yields 
the  undetermined  product  given  by 


This  product  is  neither  vector  nor  scalar  ;  it  is  known  as  &dyad. 

The  vector  or  cross  product  of  two  vectors  is  a  vector  perpen- 
dicular to  their  plane  in  the  direction  of  advance  of  a  right- 
handed  screw  when  rotated  from  the  first  to  the  second  of  these 
vectors  through  the  smaller  angle  between  them.  Its  magnitude 
is  equal  to  the  product  of  the  magnitudes  of  the  two  vectors  by 
the  sine  of  the  angle  between  them.  Therefore 

P  x  Q  =  -  Q  x  P.  (4) 

Geometrically,  this  vector  product  has  the  magnitude  of  the 
parallelogram  of  which  P  and  Q  are  the  sides,  and  a  direction 
at  right  angles  to  its  surface.  It  follows  from  simple  geometrical 
considerations  that  the  distributive  law  holds  for  this  product, 
thatis»  (?  +  Q)xR=PxR  +  QxR.  (5) 

Therefore,  inserting  crosses  between  the  vectors  in  each  term 
of  (3), 

PxQKW-^)i  +  «&-m)J  +  «<?,--W)k    (6) 

The  scalar  or  dot  product  of  two  vectors  is  a  scalar  equal  in 

magnitude  to  the  product  of  the  magnitudes  of  the  two  vec- 

tors by  the  cosine  of  the  angle  between  them.    Obviously  the 


ELEMENTS  OF  VECTOR  ANALYSIS  3 

distributive  law  holds  for   this  product.    Therefore,   inserting 
dots  between  the  vectors  in  each  term  of  (3), 


The  triple  scalar  product 

(P  x  Q)-R 

evidently  measures  the  volume  of  the  parallelepiped  of  which 
P,  Q,  and  R  are  the  edges.    Hence  the  position  of  cross  and 
dot  in  this  product  is  immaterial,  and  its  sign  is  changed  by 
interchanging  the  positions  of  two  adjacent  vectors. 
The  triple  vector  product 

(P  x  Q)  x  R 

is  obviously  a  vector  in  the  plane  of  P  and  Q.    From  simple 
geometrical  considerations  it  follows  that 

(P  x  Q)  x  P  =  P2Q  -  P-QP,  (8) 

and  (P  x  Q)  x  Q  =  P-QQ  -  Q2P.  (9) 

Now  R  may  be  written 

c(P  x  Q). 


Therefore 

(P  X  Q)  X  R  =  a(P  x  Q)  X  P  +  b(P  x  Q)  X  Q 

=  (<xP2  +  5P-Q)  Q  -  (aP-Q  +  5£2)  P 

=  P-RQ  -  Q-RP.  (10) 

This  important  expansion  may  be  put  in  words  as  follows: 
Dot  the  exterior  vector  into  the  remoter  vector  inside  the  pa- 
rentheses to  form  the  scalar  coefficient  for  the  nearer  one,  then 
dot  the  exterior  vector  into  the  nearer  vector  to  form  the  scalar 
coefficient  for  the  remoter  one,  and  subtract  this  result  from 
the  first. 

The  vector  operator  V(read  del)  is  of  great  importance  in 
mathematical  physics.  This  quantity  is  defined  as 


4  AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Let  <£  be  a  scalar  function  of  position  in  space.    Then 

V*  =  i|*  +  jf*  +  k24  (11) 

dx        %          dz 

is  known  as  the  gradient  of  </>.  It  may  easily  be  shown  to  rep- 
resent both  in  magnitude  and  direction  the  greatest  (space)  rate 
of  increase  of  <f>  at  the  point  in  question. 

Let  V=fy  +  F,j  +  r,k 

be  a  vector  function  of  position  in  space.    Then 

V.V=fi  +  P  +  ^  (12) 

dx       dy       cz 

is  known  as  the  divergence  of  V.  If  V  is  the  flux  of  a  fluid 
per  unit  time  per  unit  cross  section,  the  divergence  of  V  is  the 
excess  of  flux  out  of  a  unit  volume  over  that  into  this  volume. 
If  the  fluid  is  incompressible,  the  divergence  is  obviously  zero 
except  at  those  points  where  sources  or  sinks  are  present. 
The  vector 


is  known  as  the  curl  of  V.  If  V  specifies  the  linear  velocities 
of  the  points  of  a  rigid  body,  the  curl  is  equal  in  magnitude  and 
direction  to  twice  the  angular  velocity  of  rotation. 

The  following  identities  may  easily  be  verified  by  expansion  : 

VxV(£=0,  (14) 

V-VxV=0,  viy  (15) 

V  x  V  x  V=  VV-V-  V-VV.  (16) 

Gauss'  Theorem.  In  treating  vector  integrals  volume,  surface, 
and  line  elements  will  be  denoted  respectively  by  dr,  c?er,  and  d\. 
The  direction  of  an  element  of  a  closed  surface  will  be  taken 
as  that  of  the  outward-drawn  normal,  and  the  direction  of  an 
element  of  a  closed  curve  will  be  taken  as  that  in  which  a  right- 
handed  screw  passing  through  the  surface  bounded  by  the  curve 
must  rotate  in  order  to  advance  toward  the  positive  side  of 
this  surface. 


ELEMENTS  OF  VECTOR  ANALYSIS  5 

Let  V  be  a  vector  function  of  position  in  space.    Then  Gauss' 
theorem  states  that 


Cy.Vdr  =  Cv-dv,  (17) 

J-T  J  a 

where  the  surface  integral  is  taken  over  the  surface  <r  bounding 
the  volume  r. 

This  theorem  may  be  proved  in  the  following  way.   In  rectan- 
gular coordinates 


Let  x^  y,  z  and  #2,  ?/,  z  be  the  points  of  intersection  of  the 
surface  bounding  T  with  a  line  parallel  to  the  X  axis.    Then 

f-j^dxdydz  =  f  {Vx(xz,  y,  z)-  Vx(x^  y,  z)}  dydz 


Therefore         Cv-Vdr  =  C(Vxdydz  +  Vydzdx  +  Vzdxdy) 


Stokes'   Theorem.    If  V  is  a  vector  function   of  position   in 
space,  Stokes'  theorem  states  that 


-  fv-dX,  (18) 

J\ 


where  the  line  integral  is  taken  over  the  curve  \  bounding  the 
surface  cr. 

To  prove  this  theorem  proceed  as  follows.    In  rectangular 
coordinates 


**-£**)> 


6          AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Let  x,  y^  zx  and  x,  y^  z2  be  the  points  of  intersection  of  the 
periphery  of  cr  with  a  plane  parallel  to  the  YZ  coordinate  plane. 
Then,  taking  account  of  the  signs  of  the  differentials  involved, 


§  dz  dx  ~  5  dxdy)  = 

V,(x,  y  +  dy,  z)}dx  +  {Vx(x,  y  +  dy,  z)  -  Vx(x,  y,  z)}dx] 


fvxV-cfo^   f  (Vxdx  +  Vydy  +  Vzdz) 


Therefore 


Dyadics.  A  dyadic  is  a  sum  of  a  number  of  dyads.  The  first 
vector  in  each  dyad  is  called  the  antecedent,  and  the  second  the 
consequent.  Any  dyadic  may  be  reduced  to  the  sum  of  three 
dyads.  For  if  the  dyadic  x|/  is  given  by 

x|/  =  al  +  bm  +  en  +  do,  (19) 

the  vector  o  may  be  written 


whence         \|/  =  (a  +/d)  1  +  (b  4-  g$)  m  +  (c  +  Ad)  n.  (20) 

Similarly,  if  either  the  antecedents  or  consequents  of  a  dyadic 
are  coplanar,  the  dyadic  may  be  reduced  to  the  sum  of  two 
dyads.  Such  a  dyadic  is  said  to  be  planar.  If  either  anteced- 
ents or  consequents  are  collinear,  the  dyadic  becomes  a  single 
dyad  and  is  said  to  be  linear. 

Consider  the  dyadic 

\|/  =  al  +  bm  +  en. 
If  P  is  a  vector, 

\|;»P  =  al«P  +bm-P  +  cn^P 

is  also  a  vector.    Dotting  a  dyadic  into  a  vector,  then,  gives  rise 
to  a  vector  having  a  new  direction  and  magnitude.    This  new 


ELEMENTS  OF  VECTOR  ANALYSIS  7 

vector  is  a  linear  vector  function  of  the  original  one.  If  a  dyadic 
is  planar,  it  will  reduce  to  zero  vectors  having  a  certain  direc- 
tion, and  if  it  is  linear,  it  will  cause  all  vectors  parallel  to  a 
certain  plane  to  vanish. 

Obviously  any  dyadic  may  be  written  in  tHe  expanded  form 

\|/  =  auii  +  al2ij  +  a.13ik 


It  will  now  be  shown  that  any  dyadic  may  be  put  in  such  a 
form  that  its  antecedents  and  consequents  each  constitute  a 
right-handed  set  of  mutually  perpendicular  vectors.  Let  a  be 
a  unit  vector  of  variable  direction  extending  from  the  origin. 

Then  p  =  \|/-a 

describes  a  closed  surface  about  the  origin  as  a  varies  in  direction. 
This  surface  may  easily  be  shown  to  be  an  ellipsoid.  Let  i  be 
the  value  of  a  for  which  p  assumes  its  maximum  value  a.  Now 
consider  all  values  of  a  lying  in  the  plane  perpendicular  to  i. 
Let  j  be  the  value  of  a  in  this  plane  for  which  p  assumes  its 
greatest  value  b.  Finally,  let  k  be  a  unit  vector  perpendicular 
to  i  and  j  in  the  sense  that  will  make  i,  j,  k  a  right-handed  set. 
Let  c  be  the  value  of  p  when  a  equals  k.  Then,  as  the  dyadic 
changes  i,  j,  k  into  a,  b,  c,  it  may  be  written  in  the  form 

\|r  =  ai  +  bj  +  ck. 

Now  p  =  (ai  +  bj  +  ck)  -a, 

<Zp  =  (ai  +  bj  +  ck>da, 


When  a  is  parallel  to  i,  p  has  its  maximum  value  a,  and 
therefore  a-bj-rfa  +  a-ck-da  =  0. 

If,  moreover,  da  is  perpendicular  to  j,  a»c  vanishes,  and  if  da 
is  perpendicular  to  k,  a«b  vanishes.  Hence  a  is  perpendicular 
to  both  b  and  c. 


8  AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Now  let  a  be  restricted  to  the  jk  plane.    Then 
p-dp  =  p-bj-da  +  p-ck-da. 

When  a  is  parallel  to  j,  p  has  its  greatest  value  b,  and  therefore 
b«c  vanishes.  Therefore  a,  b,  and  c  are  mutually  perpendicular. 
If  they  do  not  form  a  right-handed  set,  the  direction  of  one  of 
them  may  be  reversed  provided  its  sign  is  changed.  Hence,  if 
it,  j\,  kj,  constitute  a  right-handed  set  of  mutually  perpendicular 
unit  vectors  parallel  respectively  to  a,  b,  c,  the  dyadic  may  be 
written  ^  =  ai^  +  ^  j  +  6,kik.  (22) 

If  \|/  =  al  +  bm  +  en, 

the  conjugate  of  \Jf  is  denned  as 


and  a  dyadic  is  said  to  be  self-conjugate  if  it    is  equal  to  its 
conjugate.    Obviously,  if  \|/  is  self-conjugate, 

x|/.P  =  P-\|/.  (23) 

The  idem/actor  I  is  defined  as 

I  =  ii  +  j  j  +  kk 


Evidently  this  dyadic  is  self-conjugate,  and  moreover 

I-P  =  P.  (24) 

It  will  now  be  shown  that  any  complete  self-conjugate  dyadic 
may  be  written  in  the  form 

x|/=<m  +  6jj  +  ekk,  (25) 

where  i,  j,  k  constitute  a  right-handed  set  of  mutually  perpen- 
dicular unit  vectors. 

It  has  already  been  shown  that  the  dyadic  may  be  written 


Now,  as  \|f  is  self-conjugate, 


ELEMENTS  OF  VECTOR  ANALYSIS 
Form  the  products 


and  \|;c-\|/  =  cfii  +  52jj  +  e2kk. 

These  are  equal,  as  the  dyadic  is  self-conjugate.    Therefore 
Put  i|/2  =  x|/.\|/c  =  \|/c.v|/. 

Now  consider  the  dyadic 

y*  -  a^l  =  (62  -  a2)  J  J,  +  (,2  -  «2)  kjq. 
Obviously  (x|/2  -  a2!)-!,  =  0. 

But,  as         x|/2  -  a2!  =  (62  -  a2)  jj  +  (c*  -  a2)  kk, 

it  follows  that  (\|/2  -  «2I)-i  =  0. 

Hence,  as  (x|;2  —  «2I)  is  planar,  but  not  linear,  ix  must  be 
parallel  to  i.    Similarly,  j\  must  be  parallel  to  j,  and  kx  to  k. 


CHAPTER  I 

THE  PRINCIPLE  OF  RELATIVITY 

1.  Motion.    The  concept  of  motion   comprises  two  essential 
factors :   a  moving  element,  and  a  reference  body  relative  to  which 
the  motion  takes  place.    A  grain  of  sand  lying  on  the  floor  of  a 
railway  car  is  not  in  motion  at  all  if  the  car  itself  is  chosen  as 
reference  body,  although  it  may  be  moving  rapidly  relative  to 
the  earth.   If,  however,  the  car,  the  earth,  and  all  other  objects 
save  the  grain  of  sand  are  removed,  the  lack  of  a  reference  body 
makes  it  impossible  to  form  a  conception  of  motion. 

A  moving  element  is  characterized  by  a  point — whether  in 
a  material  body  or  not — which  can  be  continuously  identified. 
In  the  following  discussion  a  point  always  will  be  understood 
to  have  this  property.  A  reference  body  is  essentially  a  group 
of  points  along  the  path  of  a  moving  element,  together  with  a 
device  for  assigning  numerical  values  to  the  intervals  of  time 
between  coincidences  of  the  moving  element  with  successive 
points  of  the  body,  and  to  the  distances  between  these  points. 
For  such  characteristics  are  obviously  necessary  in  order  to 
make  possible  quantitative  evaluation  of  the  motion  of  the 
moving  element. 

2.  Reference  system.    A  reference  system  is  an  assemblage  of 
points  filling   all   space.    A  device  is  provided   for  indicating 
time  at  these  points  in  such  a  way  as  to  assure  synchronism 
according  to  some  arbitrary  standard,  and  for  measuring  distances 
between  them.    This   device  is   subject  to  the  following  con- 
ditions, but  otherwise  it  is  quite  arbitrary: 

(1)  Two  points  which  are  in  synchronism  with  a  third  are 
also  in  synchronism  with  each  other. 

(2)  The  distance  between  two  points  is  independent  of  the 
time  at  which  it  is  measured. 

10 


THE  PRINCIPLE  OF  RELATIVITY  11 

Thus  a  reference  system  serves  as  a  reference  body  for  any 
moving  element.  It  must  not,  however,  be  imagined  to  offer 
any  obstruction  to  the  motion  through  it  of  such  an  element, 
or  of  another  reference  system. 

A  material  body  of  finite  extent  may  be  considered  to  consti- 
tute a  reference  system  if  the  points  of  the  body  itself  are  sup- 
posed to  have  points  outside  associated  with  them  in  such  a 
way  that  the  whole  assemblage  possesses  the  properties  described 
above.  In  order  that  the  material  part  of  such  a  system  shall  in 
no  degree  obstruct  the  motion  through  it  of  a  moving  element, 
those  portions  of  it  which  would  be  in  the  way  may  be  regarded 
as  temporarily  removed. 

The  motion  of  a  given  moving  element  may  be  described 
relative  to  an  infinite  number  of  reference  systems.  However, 
these  systems  are  not  in  general  of  the  same  significance.  For 
let  A,  B,  and  C  be  three  systems  from  which  the  motion  of  the 
moving  element  P  may  be  observed.  Suppose  it  is  found  that 
the  motion  of  P  relative  to  A  is  conditioned  by  that  of  B,  but 
is  independent  of  that  of  C.  In  such  a  case  the  motion  of  P  is 
said  to  be  related  to  B,  which  is  known  as  a  related  reference 
system.  C,  on  the  other  hand,  is  an  unrelated  or  ideal  refer- 
ence system.  Thus  for  the  motion  of  a  shot,  the  gun  from  which 
it  is  fired  constitutes  a  related  reference  system.  The  velocity  of 
a  sound  wave  is  determined,  not  by  the  motion  of  the  source, 
but  by  the  characteristics  of  the  medium  through  which  it  passes. 
Hence  in  this  case  the  source  is  an  ideal  reference  body,  while 
the  medium  is  a  related  one. 

3.  Principle  of  relativity.  In  the  case  of  light,  it  has  been 
generally  recognized,  ever  since  the  vindication  of  the  wave 
theory  by  Young  and  Fresnel,  that  the  source  does  not  consti- 
tute a  related  reference  system.  Recent  analysis  of  the  observed 
motion  of  certain  double  stars  has  confirmed  this  supposition. 
But  most  physicists  have  felt  it  necessary  to  postulate  the  exist- 
ence of  an  all-pervading  medium  in  order  to  form  a  mental 
picture  of  the  propagation  of  light  waves  through  otherwise 
empty  space.  For  a  long  time  they  were  accustomed  to  attribute 


12        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

to  this  medium,  known  as  the  ether,  the  properties  of  a  related 
reference  system.  Finally  Michelson  devised  an  experimental 
method  of  measuring  the  velocity  of  the  earth  relative  to  the 
ether,  based  on  the  assumption  that  the  ether  is  a  related  refer- 
ence system  for  the  motion  of  light.  Much  to  everyone's 
surprise,  this  velocity  turned  out  to  be  zero.  Excluding  the 
possibility  of  the  earth's  being  at  rest  relative  to  the  ether,  and 
one  or  two  other  equally  improbable  explanations,  the  only  con- 
clusion to  be  reached  was  that  the  assumption  that  the  ether  is  a 
related  reference  system  for  the  motion  of  light  was  unjustified. 
The  inference  to  be  drawn  from  the  result  of  this  experiment, 
then,  may  be  embodied  in  the  following  "  principle  of  negation." 

For  the  motion  of  an  effect  which  travels  through  empty  space, 
such  as  a  light  wave  or  one  of  the  moving  elements  which  form  an 
electromagnetic  or  a  gravitational  field,  there  is  no  related  refer- 
ence system. 

An  immediate  consequence  is  contained  in  the  following 
statement. 

If  a  law  governing  physical  phenomena  which  are  conditioned 
solely  by  those  effects  which  travel  through  empty  space,  is  deter- 
mined from  observations  made  in  two  different  reference  systems, 
the  form  of  this  law  and  the  values  of  the  constants  entering  into 
it  can  differ  in  the  two  cases  only  in  so  far  as  the  geometry  and 
devices  for  measuring  time  and  distance,  together  with  the  units 
of  these  quantities,  may  differ  in  the  two  systems.  Their  relative 
motion  in  itself  can  affect  neither  the  form  of  the  law  nor  the 
values  of  the  constants  involved.  This  is  the  principle  of  general 
relativity. 

Consider  two  reference  systems  which  have  the  same  geometry, 
devices  of  the  same  character  for  measuring  time  and  distance, 
and  interchangeable  units  of  these  quantities.  Such  systems 
may  be  said  to  be  reciprocal.  It  follows  that 

A  law  governing  physical  phenomena  which  are  conditioned 
solely  by  those  effects  which  travel  through  empty  space,  has  the 
same  form  and  its  constants  have  the  same  values  for  two  mutually 


THE  PRINCIPLE  OF  RELATIVITY  13 

reciprocal  systems.  In  the  subsequent  discussion  the  phrase 
"  principle  of  relativity "  will  be  understood  to  refer  to  this 
restricted  form  of  the  general  principle. 

4.  Reciprocal  systems.    Consider   two    reciprocal    Euclidean 
systems  S  and  $',  such  that  all  points  of  8'  have  the  same 
constant  velocity  v  relative  to  S.    Let  light  travel  in  straight 
lines   in    S  with   a  constant  speed   c.    Then   the  principle   of 
relativity  requires  that  light  shall  travel  in  straight  lines  in  S' 
with  the  same  constant  speed  c.    Let  A  and  B  be  two  points 
of  either  system  a  distance  Ar  apart.    Since  the  speed  of  light 
is  the  same  in  all  directions,  the  time  A£  taken  by  a  light  wave 
in  passing  from  A  to  B,  as  measured  in  the  system  in  which 
these  two  points  are  located,  is  the  same  as  that  taken  by  a 
light  wave  in  travelling  from  B  to  A.    Moreover, 

Ar  =  cA«.  (1) 

5.  Differential   transformations.    Let   a   set  of   right-handed 
axes  XYZ  be  fixed  in  S  so  that  the  X  axis  has  the  direction  of 
the  velocity  of  S'.    Let  a  similar  set  of  axes  X'Y'Z',  parallel 
to  XYZ  respectively,  be  fixed  in  S'.    Let  x,  y,  2,  and  t  be  the 
coordinates  of  a  point  and  the  time  at  the  point  as  measured 
in  8,  and  x\  y' ,  12',' and  t'  the  corresponding  quantities  as  meas- 
ured in  S'.    It  is  desired  to  obtain  a/,  y',  d,  and  t'  as  functions 
of  :r,  y,  2,  and  t.    Let  A'  and  B'  be  two  neighboring  points  of  S'. 
A  light  wave  leaving  A'  at  the  time  t  arrives  at  B'  at  the  time 
t  -f-  dt^  and  one  leaving  B1  at  the  same  time  t  reaches  A'  at  the 
time  t  4-  dtz,  these  times  being  measured  in  S.    If  the  coordi- 
nates of  A'  and  B'  relative  to  S  at  the  time  t  are  denoted  by 
a;,  #,  z  and  x  +  dx,  y  +  dy,  z  +  dz  respectively,  the  time  dt'  taken 
by  the  first  wave  to  travel  from  A1  to  B'  as  measured  in  S'  is 

dt'  =  d-f  (dx  +  vdtj  +  ^dy  +  ^dz  +  ^dt^ 

dx  dy  cz  dt 

and  the  equal  time  taken  by  the  second  wave  in  passing  from 
Bf  to  A1  is 

M      dt'  ^     ^         AN       W  j       dt'  *       W  j 
dt=-(-dx  +  vdt^-~dy--dz  +  -dt, 


14         AN  INTRODUCTION  TO  ELECTRODYNAMICS 
Subtracting, 

^-^}+g^  +  |*  +  ||(«,-^-0,  (2) 

and  adding, 


But  the  squares  of  the  distances  travelled  by  the  wave  going 
from  A'  to  B'  and  by  that  passing  from  B'  to  A'  as  measured 
in  S  are  respectively 

<?dtt  =  (dx  -f  vdty  +  dy2  +  rfz2, 
<fo  -  w&2+  d*+  dz2. 


Hence  ^  -  <&,  =  ^  (4) 


(5) 


where  ^r2  =  ^-h  dyz+dz\  and  v±  is  the  component  of  v  at  right 
angles  to  dr.    Substituting  this  value  of  dt^—  dt2  in  (3), 

'          dt' 


i 

1  —  p  \dx       c 


where  &  =  -• 
c 


Now  dx,  dy,  and  dz  are  arbitrary.    Hence  their  coefficients 
must  vanish,  that  is, 

-- 


Therefore  the  complete  differential  dt1  is  given  by 

,„    ^  ,  ,  a«'      ,  at'      ,  ae' 

*       "^7  *  +  T~  ^  +  T"  ^  +  ^~  dz 
dt  dx  dy  dz 


C9) 


THE  PRINCIPLE  OF  RELATIVITY  15 

To  obtain  dr',  substitute  (5)  and  (6)  in  (3).    Then 


Giving  dr  the  values  dx^  dy,  and  dz,  it  is  seen  that 

dx'  =  —  dx^  (10) 

OJ/ 

(11) 

(12) 


Now  cfa^  in  (10)  is  the  distance  of  B'  from  A'  when  the  time 
is  the  same  at  the  two  points.  If  dx  is  the  distance  of  the  position 
of  B'  at  the  time  t  +  dt  from  that  of  A'  at  the  tune  t, 

(fojj  =  dx  —  vdt. 
So  (10)  becomes 


Since  the  units  of  length  in  S  and  Sr  are  interchangeable,  it 
follows  from  symmetry  that  dy1  =  dy,  and  dz'  =  cfe.    Hence  if 

1 


k  = 


Vl-£» 

w    , 

—  =  k. 
U 

Hence  the  differential  transformations  between  the  two  sys- 
tems are 


or  dt  =  kdt' +-dx',  (13) 

dx'  =k(dx-  vdt),  dx  =  k  (dx1  +  vdt'),  (14) 

dy'=dy,  dy  =  dy',  (15) 

dz'=dz,  dz  =  dz',  (16) 


16        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

where  the  second  column  is  obtained  from  the  first  by  solving 
for  the  unprimed  differentials,  or  by  changing  the  sign  of  v. 
From  these  expressions  it  follows  that 


is  an  invariant  of  the  transformation. 

Now  the  velocity  v'  of  a  point  of  S  relative  to  Sf  is  obtained 
by  dividing  dx'  by  dt',  x  remaining  constant.    Therefore 

v'=  —  v, 

as  might  have  been  expected  from  considerations  of  symmetry. 
6.  Space  and  time  transformations.  Integrating  the  differ- 
ential relations  (13)  to  (16),  and  determining  the  constants  of 
integration  on  the  assumption  that  the  origins  of  the  two  sys- 
tems are  in  coincidence  when  the  time  at  each  is  zero, 


or     t  =  Jct'+x'  (17) 

(18) 


z'=z,  z  =  z'.  (20) 

These  four  relations  are  known  as  the  Lorentz-Einstein 
transformations. 

Consider  a  moving  element  whose  velocity  components  rela- 
tive to  S  are  Fx,  Fy,  and  Vz.  From  the  differential  relations  of 
the  preceding  section  it  follows  at  once  that 

F'=-^r.  or          K=-^-.  (21) 


(22) 


THE  PRINCIPLE  OF  RELATIVITY  17 

For  the  resultant  velocity 


Hence  if  V=  c,  V'  =  c  also,  as  should  be.    Now 


so  the  velocity  of  light  is  the  only  velocity  independent  of  v 
which  is  the  same  for  the  two  systems. 

Suppose  that  the  velocity  of  Sr  relative  to  S  is  nearly  as 
great  as  the  velocity  of  light.  Then  ft  =  1  —  8,  where  8  is  small. 
Consider  a  body  moving  in  the  X'  direction  with  a  velocity 
relative  to  S'  only  slightly  less  than  the  velocity  of  light. 
Then  F^=e(l— e),  where  e  is  small.  Equation  (21)  gives  for 
the  velocity  of  this  body  relative  to  S, 

2-8-6 
x~tf2-8-€-f8€ 


whence  Vx  is  less  than  c.  For  example,  if  v  =  0.9  c,  and  Vx  =  0.9  c, 
F"x  would  be  1.8  c  according  to  nineteenth-century  conceptions 
of  space  and  time.  But  the  addition  theorem  of  velocities  just 
obtained  from  the  principle  of  relativity  gives  Vx=  0.994  c. 

The  relations  between  components  of  acceleration  as  measured 
on  the  two  systems  are  obtained  by  differentiating  equations  (21) 
to  (23)  with  respect  to  the  time,  remembering  that 

dL_d1LdL 
dt'~dt'dt 

1  d 


18        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
These  relations  are 

f'          or       =     /j 


/,  -  ?  cw  -W          /;  +  -  cw  -/jo 

,    (25) 


/.  -  7  (/A  -w,)          f.  +  2-ain  -m 

,        f=  _  f  __  (26) 

•>*  I  v     y 


Suppose  that  the  moving  element  is  at  rest  in  /S'.  Then  the 
second  column  becomes 

/.  =  $•  (27) 

/,=•§'  (28) 

/.=§•  (29) 

7.  Four-dimensional  representation.  The  Lorentz  -Einstein 
transformations  can  be  represented  very  simply  by  a  rotation  in 
a  four-dimensional  manifold.  For  consider  a  set  of  rectangular 
axes  XYZL  in  four-dimensional  space,  such  that  the  distances 
x,  y,  z  of  a  moving  element  from  the  origin  of  S  are  measured 
along  the  first  three  axes,  and  the  quantity  l  =  ict  along  the 
fourth  axis,  where  i  =  V—  1.  The  position  of  a  moving  element 
at  a  given  time  is  represented  by  a  point  in  this  space,  and  the 
locus  of  the  positions  of  such  an  element  at  successive  instants 
by  a  line.  This  line  is  called  the  world  line  of  the  moving 
element.  Thus  the  world  line  of  a  body  permanently  at  rest 
relative  to  S  is  a  straight  line  parallel  to  the  L  axis.  The  world 


THE  PRINCIPLE  OF  RELATIVITY  19 

line  of  a  point  moving  with  velocity  V  relative  to  S  is  inclined 
to  the  L  axis  by  an  angle  <f>  such  that 

.  v 
tan  <f>  =  —  ^  —  • 

c 

Hence  the  world  line  of  a  point  of  S1  is  parallel  to  the  XL 
plane  and  makes  with  the  L  axis  an  angle  a  given  by 

tan  a  —  —  i/3,     sin  a  =  —  ik&     cos  a  =  k. 

Therefore  in  terms  of  #,  y,  2,  and  I  the  Lorentz-Einstein  trans- 
formations take  the  form 

lf  =  I  cos  a  -\-x  sin  #,     or     I  —  I1  cos  cc  —  xr  sin  a,        (30) 
xr=x  cos  a  —  Z  sin  #,  x  =  xf  cos  a  +  Z'  sin  a, 

y'=y,  y  =  y', 

2' =2,  z  =  z',  (33) 

amounting  formally  to  a  rotation  of  the  X  and  L  axes  through 
the  imaginary  angle  a. 


CHAPTER  II 

THE  RETARDED  FIELD  OF  A  POINT  CHARGE 

8.  Electric  field.  Continuous  lines  may  be  imagined  to  spread 
out  from  every  elementary  electric  charge  in  such  a  way  as  to 
diverge  uniformly  in  all  directions  when  viewed  from  the  system 
in  which,  at  the  instant  considered,  the  charge  is  at  rest.  These 
lines  are  called  lines  of  force  and,  taken 
together,  they  constitute  the  charge's  field. 
The  number  of  lines  emanating  from  an 
element  of  charge  de  will  be  supposed  to 
be  very  large,  no  matter  how  small  de 
may  be.  A  bundle  of  M  lines,  where  M 
is  a  very  large  number,  will  be  considered 
to  constitute  a  tube  of  force.  The  field 
strength,  or  electric  intensity,  E,  at  a  point 
in  a  field,  is  a  vector  having  the  direction 
of  the  lines  of  force  at  that  point  and  equal  in  magnitude  to 
the  number  of  tubes  per  unit  cross  section.  Thus  if  dN  tubes 
pass  through  a  small  surface  of  area  da-  whose  normal  makes 
an  angle  6  (Fig.  1)  with  the  field,  the  magnitude  of  the  field 
strength  is  given  by  ,«. 

da-  cos  6 
Hence  the  component  of  the  field  strength  parallel  to  dv  is 

dN 


FIG.  1 


dN 

or,  in  vector  notation,     Erfcr  =  —  d<r. 

da 

The  ratio  of  the  number  of  tubes  of  force  diverging  from  a 
charge  to  the  magnitude  of  the  charge  determines  the  unit  of 

20 


THE  RETARDED  FIELD  OF  A  POINT  CHARGE      21 

charge.  The  simplest  unit,  and  the  one  which  will  be  used  in 
the  following  pages,  is  that  which  makes  the  charge  at  any  point 
equal  to  the  number  of  tubes  of  force  diverging  from  that  point. 
This  unit  is  the  one  advocated  by  Heaviside  and  Lorentz  and, 
as  will  appear  later,  is  smaller  than  the  usual  electrostatic  unit 
by  the  factor  - 


9.  Motion  of  a  field.    Consider  an  electric  field  which  is  being 
observed  from  the  two  reference  systems  S  and  Sf  of  the  pre- 
vious  chapter.    The   principle   of   relativity  requires   that   the 
velocity  of  the  moving  elements  comprising  the  field  shall  have 
the  same  numerical  value  no  matter  whether  observations  are 
carried  on  in  S,  S'9  or  some  other  system  reciprocal  to  one  of 
these.    In  section  6  it  was  shown  that  the  velocity  of  light  is  the 
only  velocity  which  satisfies  this  condition.    Hence  the  moving 
elements  constituting  an  electric  field  must  have  the  velocity 
of  light. 

Suppose  a  charged  particle  to  be  permanently  at  rest  in  S. 
Although  the  moving  elements  constituting  its  field  are  in 
motion  with  the  velocity  of  light,  the  lines  of  force  themselves 
are  stationary.  Hence  the  motion  must  be  entirely  along  these 
lines.  Now  consider  a  charged  particle  moving  with  a  constant 
velocity  V  relative  to  S.  As  the  charge  carries  its  field  along 
with  it,  the  velocity  of  a  moving  element  will  be  along  the  lines 
of  force  only  at  points  in  the  line  of  motion.  At  all  other  points 
this  velocity  will  make  an  angle  with  the  lines  of  force  which  will 
be  greater  the  greater  the  speed.  Therefore  in  general  the  com- 
plete specification  of  an  electric  field  due  to  a  charged  particle 
requires  the  knowledge  at  every  point  of  the  values  of  two  vec- 
tors, the  field  strength  E  and  the  velocity  c.  Both  magnitude 
and  direction  of  E  must  be  given,  but  as  the  magnitude  of  c 
is  known  its  direction  only  is  required. 

10.  Transformation  equations.   Suppose  that  the  field  strength 
E  and  velocity  c  relative  to  S  are  known  for  the  field  due  to  a 
point  charge.    It  is  desired  to  find  the  values  of  these  quantities 


22        AN  INTRODUCTION  TO  ELECTRODYNAMICS 


as  measured  in  Sf.  Let  P  and  Q  (Fig.  2)  be  two  neighboring 
points  on  the  same  line  of  force,  the  coordinates  of  Q  rela- 
tive to  P  being  dx,  dy,  dz  when  the  time  is  the  same  at 
the  two  points 
according  to 
the  standards 
of  S.  Then  if 
the  time  at  Q 
should  be  later 
by  dt  than  that 
at  P,  the  coor- 
dinates of  Q 
relative  to  P 
would  become 

dx^  —  dx+cxdt, 


3»Q' 

/ 

^^'\ 

>il 

"nOr*' 

-E| 

dx_m 

.-1J-, 

w 

Vi  / 

/ 

V 

FIG. 

2 

czdt. 

Now,  in  order  that  the  times  at  P  and  Q  should  be  the 
same  when  measured  in  S1 ',  equation  (13)  of  section  5  shows 
it  to  be  necessary  that 

ft 
c      l' 

Hence  when  dt' =  0, 

dx 


THE  RETARDED  FIELD  OF  A  POINT  CHARGE      28 

Substituting  in  the  transformation  equations  (14)  to  (16), 
second  column,  of  section  5, 

dx 


are  found  to  be  the  coordinates  of  Q  relative  to  P  as  measured 
in  S'  when  the  time  is  the  same  at  the  two  points  according  to 
the  standards  of  this  system.  The  position  of  Q  at  this  time  is 
shown  by  Q'  on  the  figure.  Consequently  the  lines  of  force  in 
S1  extend  from  P  to  Q1,  instead  of  from  P  to  Q  as  in  S. 

Let  8N  be  the  number  of  tubes  of  force  passing  through  the 
area  §y§z  in  S.    Then,  since  By'  =  8y,  and  &zf  =  £z, 

$N         $N 
*~  Sy'Sz'~~          ~       ' 

-»  «-  E       E,.      E. 

Moreover,  —  -  =  —  -*-  =  -^  5 

dx      ay      dz 

K^^^K 

dx'~  dy'      dzr 


H  ence  EL  =  JcEx 


and  similarly,  E[  =  k  \  Es  +  -  (c  X  E)^  \  . 

^  G  ) 


24        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
The  magnetic  intensity  H  is  defined  by 

H--(cxE). 

c 

Therefore    E'x=  E^  (1) 

(2) 


+  w,}  =  *  {E,  +  -c  (v  x  H),}  .         (3) 


where  the  primed  and  unprimed  quantities  may  be  interchanged, 
provided  the  sign  of  v  is  changed. 

The  transformations  for  c  are  obtained  at  once  from  the  equa- 
tions (21),  (22),  and  (23)  of  section  6  for  transforming  velocity 
components.  They  are 


From  the   transformations   for   the   components   of  E  and  c 
those  for  the  components  of  H  are  readily  deduced.    For 


by  definition.  Substituting  the  values  of  the  components  of  E' 
and  c'  in  terms  of  the  unprimed  quantities  in  this  identity  and 
in  the  corresponding  expressions  for  Hy  and  Hg9  it  is  found  that 


(8) 

(9) 

A  field  formed  by  the  superposition  of  the  individual  fields 
of  a  number  of  charged  particles  is  termed  a  complex  field,  in 


THE  RETARDED  FIELD  OF  A  POINT  CHARGE      25 


contradistinction  to  the  simple  field  of  a  single  elementary  charge, 
and  the  electric  and  magnetic  intensities  in  such  a  field  are  defined 
as  the  vector  sums  of  the  corresponding  intensities  of  the  com- 
ponent simple  fields.  It  is  to  be  noted,  however,  that  the  velocity 
c,  in  so  far  as  its  direction  is  concerned,  always  refers  to  the  field 
of  an  element  of  charge,  and  never  to  the  resultant  of  a  num- 
ber of  such  simple  fields  superposed.  If  it  is  desired  to  avoid 
explicit  reference  to  the  components  of  a  complex  field,  the  field 
must  be  described  .by  means  of  equations  which  do  not  involve 
the  direction  of  motion  of  the  constituent  moving  elements. 

Since  the  transformations  that  have  been  obtained  for  E  and 
H  due  to  a  single  point  charge  are  linear  in  these  quantities, 
they  apply  as  well  to  complex  as  to  simple  fields. 

11.  Point  charge  at  rest.  Let  e  (Fig.  3)  be  a  point  charge 
momentarily  at  rest  at  the  origin  of  S  at  the  time  0.  It  is 
desired  to  find  the  QJJ 

field  strength  E  at 
P  at  a  time  t  =  r/c, 
where  r  is  the  dis- 
tance of  P  from  0. 
Two  moving  elements 
leaving  e  in  slightly 
different  directions  at 
the  time  0  will  be  at 
P  and  Q  at  the  time  t. 
At  a  time  dt,  e  will  still  be  at  0  (to  the  second  order  of  small 
quantities).  Consequently  a  moving  element  coming  from  e  at 
this  time  and  belonging  to  the  same  line  of  force  as  that  at  P 
will  reach  some  such  point  as  A  by  the  time  t.  Similarly,  one 
belonging  to  the  same  line  of  force  as  that  at  Q  will  reach  B  by 
the  time  t.  Hence,  if  rl  denotes  the  distance  OA, 


„ 

r  IG.  o 


and  rx  =  (c  +  dc)  (t  -  dt) 

=  ct  —  cdt  +  tdc, 
where  |c  +  dc|  =  |c|  necessarily,  or  dc  is  perpendicular  to  c. 


26        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
Put  p  =  r  —  rr 

Then  p  =  (cdt--dc]1 

\  c     I 

p«r  =  crdt. 

Now  since  the  lines  of  force,  as  viewed  from  S,  diverge  uni- 
formly in  all  directions  from  0  at  the  time  0,  the  number  of 
tubes  per  unit  area  passing  through  a  small  surface  at  P  with 
normal  parallel  to  r  is 


4  TIT2 

Hence  E  = 


4  Trr2  p  cos  QPH 


.  Suppose  that  the  charge  e  has  an  acceleration  f  relative  to  S. 
Then  at  the  time  dt  it  will  be  at  rest  in  some  reciprocal  sys- 
tem $',  which  has  a  velocity  f  dt  relative  to  S.  As  the  lines  of 
force  diverge  uniformly  from  the  charge  when  viewed  from  the 
system  in  which,  for  the  instant,  the  charge  is  at  rest,  and  as 
the  velocities  of  the  moving  elements  constituting  the  portions 
of  these  lines  in  the  immediate  vicinity  of  the  charge  are  along 
the  lines  themselves  when  observed  from  this  system,  it  follows 
that  if  two  moving  elements,  one  of  which  leaves  e  at  the  time  0 
and  the  other  at  the  time  dt,  are  to  lie  on  the  same  line  of  force, 
the  velocity  of  the  second  must  make  the  same  angle  in  S'  with 
the  direction  of  f  as  that  of  the  first  does  in  S.  If  the  velocities 
of  these  two  elements  are  denoted  by  c'  and  c,  and  if  the  X'  and  X 
axes  are  taken  parallel  to  f, 


But,  from  (4), 


_L  ™*~ 


THE  RETARDED  FIELD  OF  A  POINT  CHARGE 
Let  a  (Fig.  4)  be  the  angle  between  c  and  f.    Then 
cx  =  c  cos  #, 


27 


=  —  c  sin  ada. 


.       ,       fdt 
cosa  —  smada  —  - — 


Hence 


or 


cos  a  = 


da  =  —  - —  sin  a. 


Now 


dc  =  cda 

=  —fdt  sin  a, 

which  becomes,  in  vector  notation, 

(f  x  c)  x  c  ,A 
dc  =  -±  -  f  -  dt. 

C 


FIG.  4 


(11) 


Substituting  in  (10)  it  is  seen  that  if  a  charge  e  which  has  an 
acceleration  f  is  momentarily  at  rest  at  0  at  a  time  0,  the  field 
strength  at  a  point  P  distant  r  from  0  at  a  time  r/c  is  given  by 


where  the  heavy  brackets  are  used  to  denote  the  fact  that  the 
quantities  contained  therein  are  retarded;  that  is,  these  quan- 
tities refer  to  the  effective  position  of  the  charge,  or  its  position 
at  a  time  r/c  earlier  than  that  for  which  the  field  strength  at 
P  is  to  be  determined. 


Since 


H  =  -(cxE) 

c 


by  definition,       H  ^^J^-  ["_  ^{(f  x  c)  x  c}  x  ell  .  (13) 


12.  Point  charge  in  motion.  Consider  a  point  charge  e  which 
is  passing  the  origin  of  S  at  the  time  0  with  velocity  v  and 
acceleration  f  .  Choose  axes  so  that  v  is  along  the  X  axis.  Then 


28        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

this  charge  is  at  rest,  at  the  instant  considered,  in  a  system  Sf 
which  has  a  velocity  v  in  the  X  direction  relative  to  S.    So 


(ff  xc')xc'j 


are  the  values  of  E'  and  H'  at  a  point  P  distant  r1  from  0  at  the 
time  r'/c.  It  is  desired  to  determine  E  and  H  at  P  at  this  same 
instant.  Since  the  velocity  of  light  is  the  same  in  the  two  systems, 
the  time  at  P  will  be  r/c  in  S  when  it  is  r'/c  in  Sr.  Hence  the 
result  of  the  transformation  about  to  be  carried  through  will 
give  E  and  H  at  P  at  the  time  r/c. 

Let  a'  be  the  angle  in  Sf  which  the  line  OP  makes  with  the  X' 
axis.  Without  loss  of  generality  this  line  may  be  supposed  to  lie 
in  the  X'  Y'  plane.  The  Lorentz -Einstein  transformations  give 

/  cos  a'  =  Jcr  (cos  a  —  /3), 
rf  sin  a'  =  r  sin  a  ; 
whence  rr  =  Jcr  (1  —  /3  cos  a), 

.       cos  a  —  ft 
cos  a'  = 


1  —  /3  cos  a 


sin  a: 


sin  af  = 


From  (4),  (5),  and  (6>  4  =  - 


—  y8  cos  a 


4  =  0; 
and  from  (27),  (28),  and  (29)  of  Section  6, 


THE  RETARDED  FIELD  OF  A  POINT  CHARGE      29 


Now 
E=El 


X C' 


ek 


47rr'26>| 

Substituting  for  the  primed  letters  their  values  in  terms  of 
the  unprimed  ones,  and  reducing, 


or    E  = 


H  may  be  obtained  from  Hf  and  Er  in  an  exactly  analogous 
manner,  but  it  is  more  easily  found  directly  from  the  above 

expression  for  E.    For  1 

H  =  -(cxE), 

whence 


These  expressions  for  E  and  H,  it  must  be  remembered,  give 
the  values  of  these  vectors  at  P  at  the  time  r/c  in  terms  of  v 
and  f  at  0  at  the  time  0,  c  having  the  direction  of  the  line  OP. 
In  other  words,  all  the  quantities  within  the  heavy  brackets  are 
retarded.  Each  expression  consists  of  a  part  involving  the  inverse 
second  power  of  the  radius  vector  r,  and  a  part  involving  the 


30        AN  INTRODUCTION  TO  ELECTRODYNAMICS 


inverse  first  power  only.  The  latter  depends  upon  the  acceler- 
ation of  the  element  of  charge,  and  the  part  of  the  field  which 
it  determines  is  known  as  the  charge's  radiation  field. 

13.  Retarded  potentials.  In  differentiating  expressions  such 
as  those  involved  in  (14)  and  (15)  account  must  be  taken  of  the 
fact  that  the  quantities  enclosed  in  the  heavy  brackets  are 
retarded.  Let  [-^r]  be  a  scalar  whose  value  at  P  at  a  time  t  is 
given  in  terms  of  the  position  and  velocity  of  a  charged  particle 
at  0  at  a  time  t  —  r/c,  where  r  is  the  distance  OP.  Then,  if  the 
coordinates  x,  y,  z  of  P  relative  to  0  remain  unchanged, 


where,  since 


it  follows  that 


Consequently, 


dt  =  [dt]  + 


\dr] 


8t 


1- 


c-v 


dt. 


or,  symbolically, 


(16) 


On  the  other  hand,  if  x  changes  by  an  amount  dx  while  «/,  z, 
and  t  remain  constant, 


where 


[dx]  ==  dx, 


dx. 


THE  RETARDED  FIELD  OF  A  POINT  CHARGE      31 


Hence 


dx 


5? 

__£_ 

1-^1 


and  similar  expressions  hold  for  differentiation  with  respect  to 
y  and  z.    Therefore 


1- 


c.v 


(17) 


Consider  the  retarded  scalar  potential  function 


w- 


[-(-V)]' 


It  will  be  shown  that  E  and  H  due  to  a  point  charge  may  be 
expressed  as  derivatives  of  this  function  in  much  the  same  way 
as  the  field  strength  due  to  a  point  charge  in  electrostatics  may  be 
given  in  terms  of  the  gradient  of  the  potential  function  1/r.  For 


and 


-  -M- 

c  dt\_c  r  J 


Therefore 


+ 


32        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
and  comparison  with  (14)  shows  that 


Moreover, 


cxf 


lc-fj 


cxv 


-  (1  -  #2)  c  x  v  +  -3  c  x  (  {f  x  (c  -  v)}  x  c) 


and  comparison  with  (15)  shows  that 

e, 
=  4^V: 

Whether  the  field  is  due  to  a  single  point  charge  or  to  a 
number  of  such  charges,  if 


and 


a  = 


ev 


then 


} 


(18) 


(19) 


o    ' 
H  =  Vxa.  (21) 

Hence  an  electromagnetic  field  may  be  specified  by  the  values 
at  all  points  and  times  of  either  the  two  vectors  E  and  H  or 
the  scalar  potential  <f>  and  the  vector  potential  a. 


CHAPTER  III 


r 


THE  SIMULTANEOUS  FIELD  OF  A  POINT  CHARGE 

14.  Constant  velocity.  Let  a  point  charge  e  which  has  a 
constant  velocity  v  relative  to  S  be  at  the  origin  0  of  S  at  the 
time  0.  It  is  desired  to  find  the  values  of  E  and  H  at  a  point  P 
(Fig.  5)  at  the  same  instant  in  terms  of 
the  coordinates  of  P  and  the  velocity  v. 
Choose  axes  so  that  the  velocity  of  the 
charge  is  along  the  X  axis  and  the  point 
P  is  in  the  XY  plane.  The  point  Q  occu- 

T 

pied  by  the  charge  at  the  time is  its 

effective  position.  Hence  equation  (14), 
section  12,  gives  for  the  field  strength 
at  P  at  the  time  0, 


fa   \ 


o 


FIG.  5 


=  [4 


(1  —  ft  cos  a) 


5{c-v}} 


Now 


and  therefore  the  vector  c  —  v,  and  consequently  E,  have  the 
direction  OP. 

From  the  geometry  of  the  figure 

r2  =  /3V  +  E2  +  2  firR  cos  0, 


or 


Hence  -|c- v|  =  Vl- £2sin20-/3  cos0. 
Moreover, 


,*  In  order  to  avoid  cumbersome  equations  in  this  and  the  succeeding  article, 
the  retarded  quantity  r  is  not  enclosed  in  brackets. 


34        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
Therefore  E  — ^ ^ — .  • 


(1) 


The  magnetic  intensity  H  is  directed  upward  at  right  angles 
to  the  plane  of  the  figure.    Its  magnitude  is  given  by 


But 
Hence 


c  sin  (6  —  a)  = 
*    H=      e      (l-/32)/3sin0 


(2) 


The  expression  for  E  shows  that  the  lines  of  force  diverge 
radially  from  the  moving  charge  (Fig.  6),  but,  instead  of  spread- 
ing out  uniformly  in  all  directions,  as  in  the  case  of  a  static 
charge,  they  are  crowded  together  in 
the  equatorial  belt  and  spread  apart 
in  the  polar  regions.  The  greater 
the  speed  the  more  pronounced  this 
disparity,  until,  if  the  velocity  of 
light  is  attained,  the  entire  field  is 
confined  to  the  equatorial  plane. 

If  lines  are  drawn  so  as  to  have 
everywhere  the  direction  of  the  vector 
H,  these  magnetic  lines  of  force  will 
be  circles  in  planes  at  right  angles 
to  the  line  of  motion  with  centers  lying  on  this  line.  If  the 
magnetic  lines,  like  the  electric  lines,  indicate  by  their  density 
the  magnitude  of  the  corresponding  vector,  a  similar  crowding 
together  in  the  equatorial  belt  and  spreading  apart  in  the  polar 
regions  will  exist.  However,  the  total  number  of  magnetic  lines 
of  force  in  the  field,  unlike  that  of  the  electric  type,  will  be 
greater  the  greater  the  speed  of  the  charged  particle. 

15.  Constant  acceleration.  Consider  a  point  charge  moving 
with  an  acceleration  <)>,  which  always  has  the  same  value  relative 
to  that  system,  reciprocal  to  $,  in  which  the  charge  happens  to 
be  at  rest  at  the  instant  Considered.  Let  this  charge  come  to 
rest  momentarily  at  the  origin  0  of  S  at  the  time  0.  It  is  desired 


FIG.  6 


SIMULTANEOUS  FIELD  OF  A  POINT  CHARGE      35 


to  find  the  values  of  E  and  H  at  a  point  P  (Fig.  7)  at  the  same 
instant  in  terms  of  the  coordinates  of  P  and  the  acceleration  <(>. 
Choose  axes  so  that  <|>  is  along  the  X  axis  and  P  lies  in  the 
XY  plane.  The  effective  position  of  the  y 
charge  is  the  point  Q,  which  it  occupied 

at  the  time  —  -  •    From  (27),  section  6, 


Integrating,  the  velocity  at  Q  is  found 
to  be  given  by 


and  the  distance  OQ  by 


0 


Q 
FIG.  7 


If  the  components  of  E  along  and  at  right  angles  to  QP  are 
determined  separately,  it  follows  from  equation  (14),  section  12, 
that 

1          "•  (3) 


showing  that  E  is  parallel  to  c. 
Now,  from  geometry, 


•^=-cos<9  + 


Ecos0=rcosa+  OQ. 


Hence      l+/3cosa  = 


1  + 


36         AN  INTRODUCTION   TO  ELECTRODYNAMICS 

Substituting  in  the  expression  for  Er, 
e  1 


(4) 


To  find  the  components  of  E  along  and  perpendicular  to  OP 
it  is  necessary  to  obtain  the  values  of  cos  8  and  sin  8.  From  the 
geometry  of  the  figure  it  follows  that 


cos    = 


sn    = 


LT 

\1  + 

> 


whence 


(5) 


(6) 


To  obtain  the  equation  of  the  lines  of  force,  use  may  be  made 

of  the  relation  ,/, 

^ 

dB 


The  solution  of  this  differential  equation  is 


e  sin 


SIMULTANEOUS  FIELD  OF  A  POINT  CHARGE       37 


where  b  is  the  constant  of  integration.    Writing  this  equation  in 


\ 


(7) 


it  is  seen  that  the  lines  of  force  are  circles  passing  through  0 
with  centers  in  a  plane  at  right  angles  to  the  X  axis  and  at 
a  distance  — <?2/<£  from  0.  Fig.  8  shows  the  section  of  the  field 
cut  by  a  plane  through  the 
X  axis.  The  full  lines  repre- 
sent the  lines  of  force. 

Consider  an  electric  field 
all  parts  of  which  happen  to 
be  at  rest  in  a  single  system 
S  at  the  same  time.  A  sur- 
face in  $,  so  constructed  as  to 
be  everywhere  normal  to  the 
lines  of  force,  is  called  a  level 
surface.  For  a  field  perma- 
nently at  rest  in  S  level  surfaces  are  identical  with  the  equipo- 
tential  surfaces  of  electrostatics.  In  the  case  under  discussion  of 
a  charged  particle  moving  with  constant  acceleration,  E  has  been 
shown  to  be  parallel  to  c  simultaneously  at  all  points.  Hence  all 
parts  of  the  field  are  at  rest  at  the  same  time  and  consequently 
level  surfaces  can  be  constructed.  The  differential  equation  of 
these  surfaces  is,  dQ 

dE  = 

1  + 


FIG.  8 


ip 
cos* 


of  which  the  solution  is 


6'2 


(8) 


where  h  is  the  constant  of  integration.  This  is  the  equation  of 
a  family  of  spheres  with  centers  at  the  effective  positions  of  the 
moving  charge.  Their  traces  are  shown  by  broken  lines  in  Fig.  8. 
Since  E  is  parallel  to  c  at  the  instant  considered,  H  is  every- 
where zero. 


38         AN  INTRODUCTION  TO  ELECTRODYNAMICS 

16.  General  case.  The  retarded  expressions  for  E  and  H 
deduced  in  the  preceding  chapter  show  that  the  field  at  a  point 
P  and  time  0  is  conditioned  by  the  motion  of  the  charge  pro- 

\T\ 

ducing  that  field  at  a  time  —  —  >  where  [r]  is  the  distance  of  P 

from  the  effective  position  of  the  moving  particle.  Therefore 
the  specification  of  the  entire  field  at  the  time  0  involves  the 
complete  past  history  of  the  charged  particle.  Since,  for  physical 
reasons,  the  motion  of  this  particle  must  be  continuous,  the  past 
history  of  its  motion  is  contained  in  its  present  position,  veloc- 
ity, acceleration,  and  higher  time  derivatives  of  the  positional 
vector.  Hence  the  simultaneous  values  of  E  and  H  may  be 
expressed  as  series  in  these  quantities.  While  these  series  may 
fail  to  converge  for  distant  portions  of  the  field,  or  for  very 
rapidly  changing  motion,  their  form  will  make  evident  their  very 
rapid  convergence  for  all  cases  to  which  they  will  be  applied. 

If  the  point  at  which  E  and  H  are  to  be  evaluated  is  chosen 
as  origin,  and  if  [a;],  [?/],  [z]  are  the  coordinates  of  the  effec- 
tive position  of  the  charged  particle  relative  to  this  origin, 


etc.,   and  the  x  component  of  the  retarded  expression  (14), 
section  12,  for  the  field  strength,  may  be  written 


r     c/ 

If  x,  y,  z  are  the  coordinates  of  the  simultaneous  position  of  the 
charge,  and  v,  f  ,  f  •  •  •  its  simultaneous  velocity,  acceleration,  etc., 


m=/-/      +  /- 

iJxl      J  x      J  x         i   9          2        R          3          ' 
and  similar  expressions  hold  for  the  y  and  z  components.    Put 


=  -  -  _  _ 

*~~S     =7'  7*=  7"'  z  =  T'  *= 


SIMULTANEOUS  FIELD  OF  A  POINT  CHAKGE      39 

Then 
=  ,2l_  2    -mM  +  (V*  +  £2)  £    - (8.m 


12 


60  v 

Now  put  T  ==  -^ 


e  =  (J-m  +  5  €-p  +  10  8-y)  Ar5, 
whence  the  previous  equation  becomes  simply 


J 

~&V' 

After  some  reduction  it  is  found  that 


+  0 


40        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Hence  Ex  =  -  — —„  krl~  *J.  (11) 

4-Trr2 

Returning  to  (10)  and  solving  for  r  by  successive  approxi- 
mations, it  is  found  that 


....     (12) 

Substituting  this  value  of  r  in  (11)  and  reducing,  the  fol- 
lowing expression  is  obtained  for  the  X  component  of  the 
electric  intensity, 

ek    f      /.,      3    o     15   4  1 ,      9    27 

JSX  =  -  -A -A  mr(  1-  -  «2+  —  a4 .  .  .  +  -  b  -  -a*l  .  . . 

4 


The  X  component  of  the  magnetic  intensity  may  be  obtained 
most  easily  from  the  relation 


which  gives,  after  some  reduction, 


-  tf(S,mt  -  S.OT,)  +  - 


Equations  (13)  and  (14)  give  E  and  H  at  the  origin  in  terms 
of  the  simultaneous  coordinates,  velocity,  acceleration,  etc.  of 
the  charge  e.  To  obtain  the  values  of  E  and  H  at  a  point  #,  «/,  z 


SIMULTANEOUS  FIELD  OF  A  POINT  CHARGE      41 

due  to  a  charge  e  at  the  origin,  the  signs  of  the  coordinates  in 
these  two  equations  must  be  changed,  giving 


where 


CHAPTER  IV 

THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON 

17.  Electrical  theory  of  matter.  All  matter  will  be  assumed 
to  be  made  up  of  positive  and  negative  electrons.  An  electron 
will  be  defined  as  an  invariable  charge,  of  magnitude  approxi- 


Heaviside-Lorentz  units,  distributed  over  a  surface  which  is 
spherical  in  form  to  an  observer  in  that  system,  reciprocal  to  S, 
in  which  the  electron  happens  to  be  momentarily  at  rest.  A 
positive  electron  will  be  considered  to  differ  from  a  negative  one 
only  in  the  sign  of  the  charge  involved  and  the  radius  of  the 
spherical  surface  over  which  it  is  distributed. 

The  electromagnetic  force  dK.  on  an  element  of  charge  de,  as 
measured  in  that  system,  reciprocal  to  S,  in  which  this  charge 
happens  to  be  at  rest  at  the  instant  considered,  is  denned  as  the 
product  of  the  field  strength  E  by  the  charge  de.  The  extension 
of  this  definition  to  the  case  of  a  system  in  which  de  is  not  at 
rest,  will  be  given  in  the  next  section. 

The  distribution  of  charge  on  the  surface  of  an  electron  will 
be  supposed  to  be  such  as  to  make  the  tangential  force  due  to 
its  own  field  zero  at  all  points  of  the  surface,  all  measurements 
being  made  in  that  system  relative  to  which  the  electron  is 
momentarily  at  rest.  This  assumption  is  introduced  merely  for 
the  purpose  of  simplifying  the  analysis  (section  20)  involved  in 
determining  the  dynamical  equation  of  an  electron  moving  with 
constant  acceleration.  To  the  number  of  terms  to  which  the 
analysis  is  carried  in  the  general  case  (section  21),  no  change 
in  the  dynamical  equation  is  introduced  if  this  hypothesis  is 
replaced  by  the  more  probable  assumption  that  the  distribution 
of  charge  is  such  as  to  make  the  tangential  force  due  to  the 

42 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON    43 

total  field,  that  is,  the  resultant  of  the  impressed  field  and  the 
electron's  field,  equal  to  zero,  or  by  the  simple  assumption  that 
the  distribution  of  charge  is  always  uniform. 

18.  Dynamical  assumption.  In  the  previous  chapters  the 
discussion  has  been  concerned  with  the  determination  of  the 
field  of  a  charged  particle.  It  must  be  borne  in  mind,  however, 
that  the  lines  of  force  constituting  such  a  field  are  nothing 
more  than  convenient  geometrical  representations  to  be  em- 
ployed in  describing  the  effect  of  one  charged  particle  on 
another,  and  that  no  reason  exists  for  attributing  a  greater 
substantiality  to  them  than  to  any  other  arbitrary  convention, 
such  as,  for  instance,  parallels  of  latitude  on  the  earth's  sur- 
face. The  representation  of  a  field  by  lines  of  force  has  led  to 
the  concept  of  electric  intensity,  and  the  electromagnetic  force 
on  an  element  of  charge,  as  measured  in  the  system  in  which 
the  charge  is  momentarily  at  rest,  has  been  defined  in  terms 
of  this  quantity.  In  order  to  pass  from  these  definitions  to 
the  quantitative  description  of  the  effect  of  one  electron  on 
another,  it  is  necessary  to  introduce  the  following  dynamical 
assumption : 

The  motion  of  an  electron  is  such  as  to  make  the  total  electro- 
magnetic force  on  it,  as  measured  in  that  system,  reciprocal  to  S, 
in  which  it  happens  to  be  momentarily  at  rest,  equal  to  zero. 
By  the  total  electromagnetic  force  is  to  be  understood  the 
resultant  of  the  force  due  to  the  impressed  field  and  that  due 
to  the  charge's  own  field.  With  forces  which  are  not  electrical 
in  nature,  such  as  must  exist  if  a  dynamical  explanation  of  the 
stability  of  the  electron  is  possible,  the  present  discussion  is 
not  concerned.  While  extra-electrical  stresses  on  a  single  elec- 
tron may  be  of  great  intensity,  their  resultant  will  be  assumed 
to  be  always  zero.  Moreover,  such  forces  will  be  supposed  to 
be  comparatively  negligible  when  the  effect  of  one  electron  on 
another  is  under  consideration.  Thus  no  account  will  be  taken 
of  the  gravitational  attraction  between  two  electrons,  as  it  will  be 
deemed  quite  unimportant  compared  to  the  electrical  attraction 
or  repulsion. 


44        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Consider  an  electron  which  is  at  rest  in  Sr  at  the  instant 
considered.  Then  the  dynamical  assumption  stated  above  re- 
quires that  /» 

E'de  =  0.  (1) 

Substituting  for  the  components  of  Er  their  values  in  terms 
of  the  components  of  E  and  H,  as  given  in  equations  (1),  (2),  (3), 
section  10,  it  is  found  that 

Cflxde  =  0, 


Hence  it  is  natural  to  extend  the  definition  of  electromagnetic 
force  given  in  the  preceding  section  so  as  to  read : 

The  electromagnetic  force  c?K  on  an  element  of  charge  de,  as 
measured  in  a  system,  reciprocal  to  S,  relative  to  which  the 
charge  has  the  velocity  v  at  the  instant  considered,  is  defined  by 

.  (2) 

Then  the  dynamical  assumption  may  be  stated  in  the  more 
general  form : 

The  motion  of  an  electron  is  such  as  to  make  the  total  electro- 
magnetic force  on  it,  as  measured  in  any  system  reciprocal  to  S, 
equal  to  zero.  Thus  the  dynamical  equation  of  an  electron  may 
be  found  directly  for  any  system,  no  matter  whether  the  elec- 
tron is  at  rest  in  that  system  or  in  motion  with  respect  to  it. 
However,  in  order  to  avoid  unnecessary  analysis,  the  method 
pursued  will  be  first  to  deduce  this  equation  relative  to  that 
system  in  which  the  electron  is  momentarily  at  rest,  and  then 
to  extend  it  to  other  systems  by  means  of  the  transformations 
already  derived. 

19.  Constant  velocity.  Consider  an  electron  permanently  at 
rest  in  S'.  Relative  to  an  observer  in  this  system  the  electron  is 
a  uniformly  charged  spherical  surface  of  radius  a  with  a  uniform 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON    45 

radial  field.  To  an  observer  in  $,  however,  this  electron  has 
a  constant  velocity  v  along  the  X  axis,  and  the  transforma- 
tion equation  (18),  section  6,  shows  that  its  dimensions  in  the 
direction  of  motion  are  shorter  in  the  ratio  1  :  k  when  viewed 
from  this  system,  while  those  at  right  angles  to  this  direction 
are  unchanged.  Hence  to  an  observer  relative  to  whom  an 
electron  is  moving  its  surface  is  that  of  an  oblate  spheroid  with 
the  short  axis  in  the  direction  of  motion. 

Describe  two  right  circular  cones  with  vertices  at  the  center  0' 
of  the  electron  and  axes  along  the  X  axis  such  that  elements  of 
the  cones  make  angles  0'  and  0'  +  dd'  respectively  with  their 
common  axis.  If  e  is  the  charge  on  the  electron,  the  number  of 
tubes  included  between  the  cones  is 


M  COS0 

But  cosfl'  ==  - 

Vl-/32sin20 


22 


(l-/32sin 


e  (- 
Hence  dN=--  -  * 

2  (l-/32sin20)* 
and  the  electric  intensity  in  S  at  any  distance  R  from  0'  is 

'      '  fl"*8)       .  (3) 


The  magnetic  intensity  is  given  by 

H=-\cxE\ 

c 


and  substituting  for  E  its  value  from  (3), 

e        l- 


46        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Comparison  of  these  expressions  with  (1)  and  (2),  section  14, 
shows  that  the  external  field  of  an  electron  moving  with  con- 
stant velocity  is  the  same  as  that  of  an  equal  charge  located  at 
its  center.  It  follows  from  symmetry  that  the  resultant  force 
on  the  electron  due  to  its  own  field  is  zero.  Hence  the  dynamical 
assumption  requires  that  the  impressed  force  shall  be  zero  as  well. 

20.  Constant  acceleration.  Consider  an  electron  each  point  of 
which  moves  with  an  acceleration  which  always  has  the  same 
value  relative  to  that  system,  reciprocal  to  S,  in  which  this  point 
happens  to  be  at  rest  at  the  instant  considered.  Let  <|>  be  the 
value  of  this  acceleration  for  the  point  0  of  the  electron.  Choose 
axes  so  that  <|>  is  along  the  X  axis.  Then  (27),  section  6,  gives 
for  the  acceleration  /  of  this  point  relative  to  S  at  any  time 

-  /=^,(l-^)i  .     '  (5) 

Integrating,  the  velocity  of  0  relative  to  S  is  found  to  be 


(«) 

4+*? 

and  the  displacement  by 


Consider  a  neighboring  point  P  of  the  electron  such  that  OP 
is  parallel  to  the  X  axis  and  equal  to  d\  when  0  is  at  rest  in  S. 
Then,  since  0  and  P  have  constant  accelerations  relative  to  the 
systems,  reciprocal  to  $,  in  which  they  happen  to  be  at  rest  at 
the  instant  considered,  the  principle  of  relativity  requires  that 
the  length  OP  as  measured  by  an  observer  in  any  system  recip- 
rocal to  S  when  O  is  at  rest  in  that  system  shall  be  the  same  as 
the  length  OP  as  measured  by  an  observer  in  S  when  0  is  at 
resting.  Hence  dp  =  d\^T^i?  (8) 

is  the  distance  OP  as  measured  in  S  when  0  has  the  velocity  @c. 

Now,  when  0  is  at  rest  in  $,  P  may  have  a  velocity  dp  in 

the  X  direction.    But  the  principle  of  relativity  requires  that  dp 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON    47 

shall  be  the  same  relative  to  any  other  system,  reciprocal  to  S, 
at  the  instant  when  0  happens  to  be  at  rest  in  that  system. 
Hence,  adding  to  d\  the  difference  between  the  displacements 
of  P  and  0  in  the  time  t, 

oe  =  d\  +  $!£{i-JT=&}+fi&  (9) 

9  9 

is  found  to  be  the  distance  OP  as  measured  in  S  when  0  has 
the  velocity  fie. 

.  Equating  coefficients   of  like  powers   of  @  in  the  identical 
expressions  (8)  and  (9),  it  is  seen  that 


The  first  of  these  equations  shows  that  when  one  point  of  the 
electron  is  at  rest  in  S,  every  other  point  is  likewise  at  rest.  Inte- 
grating the  second, 

(10) 


where  <£0  is  the  acceleration  of  0.  This  equation  shows  that  points 
on  the  forward  side  of  the  electron  have  smaller  accelerations 
than  those  on  the  rear.  Such  a  difference  is  obviously  necessary 
in  order  to  produce  the  progressive  contraction  of  the  electron 
required  by  the  principle  of  relativity  as  its  velocity  relative 
to  S  increases. 

Obviously,  the  relations  just  obtained  between  the  velocities 
and  accelerations  of  points  of  the  electron  under  consider- 
ation apply  equally  well  to  points  of  the  field  of  Fig.  8,  p.  37. 
Hence  any  one  of  the  level  surfaces  of  this  field,  such  as  that 
upon  which  the  point  P  lies,  may  be  considered  to  constitute 
the  surface  of  the  electron.  As  the  charge  is  distributed  entirely 
on  this  surface,  it  is  necessary,  in  order  that  the  external  field 
should  be  the  same  as  that  due  to  an  equal  charge  at  0,  that 
the  density  of  charge  should  be  everywhere  equal  to  the  electric 
intensity  just  outside  this  surface. 


48        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

If  e  is  the  electron's  charge,  and  E±  the  strength  of  the  external 
field,  the  impressed  force  is 


In  Fig.  8,  Q  is  the  geometrical  center  of  the  electron  and  a 
its  radius.  The  electric  intensity  EZ  at  its  surface  due  to  an 
equal  charge  at  0  is  given  by 

e  1 


where 


$0  being  the  acceleration  of  the  point  0.  As  the  field  due  to  the 
electron  vanishes  everywhere  within  its  interior,  the  resultant 
force  K2  on  this  charged  particle  due  to  its  own  field  is 

2  cos  ado-j 

where  do-  is  an  element  of  the  surface.  Substituting  for  E  its 
value  in  terms  of  a  and  integrating, 


The  point  on  the  axis  of  symmetry  of  the  electron  through 
which  a  perpendicular  plane  would  divide  its  surface  into  parts 
having  equal  charges,  will  be  called  the  center  of  charge.  If  </> 
is  the  acceleration  of  this  point, 


Hence 


The  rest  mass  m  is  defined  by 

e2 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON    49 
Hence,  as  the  dynamical  assumption  requires  that 


it  follows  that  the  acceleration  of  the  center  of  charge  is  deter- 
mined by  eE,  =  mf  (15) 

at  the  instant  that  the  electron  is  at  rest  in  S. 

Consider  an  electron  which  has  the  type  of  motion  under 
discussion,  and  which  is  at  rest  in  S1  at  the  instant  considered. 
Let  the  acceleration  f  of  the  center  of  charge  make  an  angle 
with  the  direction  of  $"s  velocity  relative  to  S.  Then,  dropping 
the  subscript,  ^ 


Substituting  in  each  of  the  component  equations  the  values 
of  JE£,  E'y,  E[  from  (1),  (2),  (3),  section  10,  and  those  of  /J, 
/;,  /;  from  (27),  (28),  (29),  section  6,  it  is  found  that 

eEx=mtffx,  (16) 


^/2.  (18) 

As  the  electromagnetic  force  in  S  is  denned  by 


it  is  seen  to  be  necessary  to  distinguish  between  the  longitudinal 

^-inx 

(19) 


and  the  transverse  mass  ,  ,OAX 

mt  =  mk.  (20) 

Both  masses  increase  with  the  velocity,  becoming  infinite  as 
the  velocity  of  light  is  approached.  In  terms  of  the  transverse 
mass  the  dynamical  equation  may  be  written  in  the  compact  form 


(21) 
where  K  is  the  impressed  force. 


50        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

21.  General  case.  Consider  an  electron  a  point  P  of  which 
is  at  rest  in  S  at  the  time  0.  Denote  by  f,  f,  etc.  the  acceleration, 
rate  of  change  of  acceleration,  etc.  of  this  point.  Choose  axes 
so  that  the  X  axis  has  the  direction  of  f.  Then  if  Q  is  a  neigh- 
boring point  of  the  electron  whose  coordinates  relative  to  P  are 
dx,  dy,  dz  at  the  time  0,  the  values  of  these  coordinates  at  the 
time  dt  will  be 

dxt  =  dx  +  (-^dx  +  d^Ldy  +  d^  dz]  dt 
\dx  dy  dz      / 


and    similar    expressions    for    dyt    and    dzt.     But,    as   /x=/, 


-i^2..\ 

dyt  =  dy, 
dzt  =  dz. 

Equating  coefficients  of  like  powers  of  dt  and  dx,  dy,  dz  in 
the  equivalent  expressions  for  dxt,  dyt,  dzt,  it  is  found  that 

dvx      dvx      dvx      dVy  s\ 

dx       dy       dz       dx 


Hence  the  velocity  is  not  a  function  of  the  coordinates,  and  when 
one  point  of  the  electron  is  at  rest  in  S,  all  other  points  are  also 
at  rest.  Moreover,  the  y  and  z  components  of  the  acceleration 
are  not  functions  of  the  coordinates,  and  the  x  component  is  a 
function  of  x  only. 

Before  proceeding  further,    it  is  convenient  to   distinguish 
between  the  orders  of  possible  factors  which  may  appear  in  the 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON    51 

dynamical  equation  of  the  electron.    If  a  denotes  the  radius  of 
the  electron,  it  will  be  considered  that 

/3  is  of  the  first  order, 

^  is  of  the  second  order, 

f   2 

^-  is  of  the  third  order, 

•/•**  B 
i-J-  is  of  the  fourth  order,  and 

c 

'-f'$ 

^  is  of  the  fifth  order. 

The  dynamical  equation  to  be  obtained  will  be  carried  through 
the  fifth  order  as  thus  defined. 

As  before,  the  impressed  force  on  the  electron  is 

K1=eE1.  (23) 

The  next  step  is  to  evaluate  the  reaction  on  the  electron  of 
its  own  field.  Let  the  origin  be  located  at  a  point  0  on  the 
surface  of  the  electron,  and  for  the  purposes  of  the  following 
analysis  let  the  orientation  of  the  axes  relative  to  f  be  arbitrary. 
Then  if  P  is  another  point  on  the  surface  of  the  electron  whose 
coordinates  relative  to  0  are  x,  y,  z,  equation  (15),  section  16, 
gives  for  the  force  exerted  by  a  charge  de  at  0  on  a  charge  del  at  P 
dede.  f  /,  1  .3 2.1  3  , 


4  7TT2    I 


f      /-,     1 
\m*(  1~2 


.          <24> 

Integration  of  this  expression  with  respect  to  del  will  give  the 
force  exerted  on  the  rest  of  the  electron  by  the  charge  de  at  0. 
Finally,  on  integrating  with  respect  to  de,  the  X  component  of 
the  force  on  the  electron  due  to  the  reaction  of  its  own  field 


52        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

will  be  obtained.  In  performing  these  integrations,  the  charge 
on  the  electron  may  be  considered  to  be  uniformly  distributed 
over  its  surface,  for,  even  under  the  conditions  assumed  in  the 
last  section,  reference  to  (12)  shows  that  the  divergence  from 
uniformity  there  implied  leads  to  no  term  of  less  than  sixth 
order  which  does  not  vanish  upon  integration.  Moreover,  it  is 
unnecessary  to  take  into  account  the  variation  of  f  from  point 
to  point  of  the  electron,  since  (22)  shows  that  the  only  term 
involved  of  less  than  sixth  order  vanishes  when  the  integration 
is  performed.  A  fortiori  the  variations  of  the  derivatives  of  f  are 
negligible. 

Omitting  from  the  integrand  all  terms  which  vanish  on  inte- 
gration, (21)  leads  to  the  expression 


Since 


™2  Qm  + 1 

—  rmdede,  = aV, 

M*  1  /VT1  I  *7 


e  e      f        ea    f  j  ^ 

2/'+*  /-+J- 


and  similar  expressions  hold  for  the  y  and  z  components.    Hence 
the  dynamical  assumption  leads  to  the  vector  equation 


Consider    an   electron   which   is   momentarily  at  rest  in   S'. 
Then,  dropping  the  subscript, 


67mC2  67TC3         ""QTTC'4  187TC5 

Equations  (27),  (28),  (29),  section  6,  give 


f 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTION    53 

Differentiating  (24),  (25),  (26),  section  6,  once,  twice,  and 
thrice,  with  respect  to  tf,  and  then  placing  V  equal  to  zero,  it 
is  found  that 


:£, 


f'x  =  k?fx+  terms  of  sixth  and  higher  orders, 

#  -*£+•••, 
/:=*£+•••; 
/]=*'/,'+•••, 


Substituting  in  each  component  of  the  dynamical  equation 
these  values  of  /j,  /J,  /^,  and  their  derivatives,  as  well  as  the 
values  of  E'x,  E^  Ez,  from  (1),  (2),  (3),  section  10,  the  dynam- 
ical equation  of  an  electron  for  a  system  relative  to  which  it 
has  a  velocity  v  is  found  to  have  the  following  components: 

027-3  02J.fi 

eE  =  f  --l^-f-Bf  ... 

' 


1TA 

e  /c 


187T6'5 


18 


54        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

These  equations  show  that  it  is  necessary  to  distinguish 
between  the  coefficients  of  the  longitudinal  and  transverse 
components  of  the  acceleration  and  of  each  of  its  derivatives. 
All  the  coefficients  approach  infinity  as  the  velocity  of  the 
electron  approaches  that  of  light,  and  the  series  cease  to 
converge. 

Including  all  terms  not  higher  than  the  third  order,  the 
dynamical  equation  of  an  electron  may  be  put  in  the  form 

K  =  wf-rcf...,  (30) 

where  K  =  e  JE  +  -  (v  X  H)| 

is  the  impressed  force,  and 

e* 


6 


22.  Rigid  body.  Consider  an  element  of  volume  dr  of  a 
material  body  large  enough  to  contain  a  vast  number  of  positive 
and  negative  electrons,  but  small  compared  to  the  total  volume 
of  the  body.  A  rigid  body  will  be  defined  as  one  all  such 
elements  of  which  maintain  the  same  relative  configuration  and, 
on  the  average,  the  same  internal  constitution  with  respect  to 
the  system  in  which  the  body  is  momentarily  at  rest,  whatever 
external  conditions  the  body  may  be  subject  to. 

Consider  a  rigid  body  momentarily  at  rest  in  system  S. 
Equations  (27),  (28),  (29)  give  for  the  dynamical  equation  of 
an  electron  in  this  body 

-^..,  (31) 

through  the  third  order,  which  is  as  far  as  the  analysis  will  be 
carried  in  this  section. 

The  electric  and  magnetic  intensities  appearing  in  this  expres- 
sion may  be  separated  into  the  intensities  Ee  and  He  due  to  the 
impressed  field,  and  the  intensities  E0  and  H0  due  to  the  fields 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON    55 

of  the  other  electrons  in  the  rigid  body.    For  the  latter,  equa- 
tions (13)  and  (14),  section  16,  give 

ek 


where  the  summation  extends  over  all  the  electrons  except  the 
one  under  consideration. 

Suppose  now  that  there  is  no  external  field  and  that  the 
rigid  body  is  permanently  at  rest  in  S.  Symmetry  requires  that 
as  many  electrons  in  any  element  of  volume  dr  shall  have  a 
given  velocity  or  acceleration  in  one  direction  due  to  the  internal 
motion  as  in  any  other  direction.  This  same  condition  must  be 
satisfied  in  the  presence  of  an  impressed  field,  for  the  internal 
constitution  of  a  rigid  body  is  independent,  by  definition,  of 
the  external  conditions  to  which  it  may  be  subject. 

To  return  to  the  case  of  a  rigid  body  momentarily  at  rest  in 
S  in  the  presence  of  an  impressed  field,  let  fe  and  fe  be  the 
acceleration  and  rate  of  change  of  acceleration  of  the  body  as 
a  whole,  and  v0,  f0,  and  f0  the  velocity,  acceleration,  and  rate  of 
change  of  acceleration  of  an  electron  of  the  body  due  to  the 
internal  motion.  Summing  up  over  all  the  electrons, 


and  therefore 


(32) 


where  the  double  summation  is  for  all  values  of  both  i  and  j 
such  that  .  . 


56         AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Consider  a  rigid  body  at  rest  in  Sf.    Then,  dropping  the 
subscript  e, 


Substituting  for  Ef  its  value  in  terms  of  E  and  H,  and  for  f  ' 
and  f  then:  values  in  terms  of  f  and  f,  the  dynamical  equation 
of  a  rigid  body  takes  the  form 


The  first  term  in  the  brace  multiplying  f  is  the  sum  of  the 
masses  of  the  electrons  composing  the  rigid  body,  while  the 
second  term  is  the  sum  of  the  mutual  masses  of  these  electrons. 
While  the  mass  of  an  electron  must  be  positive,  the  mutual 
mass  of  two  may  be  positive  or  negative  according  as  they  have 
like  or  unlike  signs.  Hence  the  mass  of  a  rigid  body  is  greater 
the  more  electrons  there  are  of  the  same  sign.  The  same  is  true 
of  the  coefficient  of  the  rate  of  change  of  acceleration.  In  fact 
this  coefficient  vanishes  if  the  body  is  uncharged. 

Denoting  by  K  the  impressed  force,  and  by  m  and  n  the  con- 
stant coefficients  of  f  and  f  respectively,  the  dynamical  equation 
of  a  rigid  body  takes  the  form 

K  =  mi  -  nf  •  •  ..  (34) 

A  conductor  carrying  a  current  may  be  considered,  in  so  far 
as  the  expressions  arrived  at  in  this  section  are  concerned,  as  one 
rigid  body  through  which  another  is  passing.  As  the  electrons 
carrying  the  current  are  all  of  the  same  sign,  their  mutual  masses 
are  positive,  and  the  mass  of  the  current  is  greater  than  the  sum 
of  the  masses  of  the  individual  electrons  which  constitute  it. 

23.  Experimental  determination  of  charge  and  mass  of  electron. 
Consider  a  stream  of  negative  electrons  from  the  cathode  of  a 
discharge  tube  travelling  at  right  angles  to  the  lines  of  force  of 


THE  DYNAMICAL  EQUATION  OF  AN  ELECTRON    57 

mutually  perpendicular  electric  and  magnetic  fields.    The  electro- 
magnetic force  on  each  electron  is  given  by 


Hence  if  -vxH  =  — E, 

c 

this  force  will  vanish,  and 

E 


By  adjusting  crossed  electric  and  magnetic  fields  so  as  to 
produce  no  deflection  in  a  beam  of  cathode  rays,  J.  J.  Thomson 
has  found  the  velocity  of  these  charged  particles  to  be  about 
one-tenth  that  of  light. 

If,  now,  the  magnetic  field  be  suppressed, 

K  =  eE, 

and  to  a  first  approximation 

mi  =  eE. 
If  the   rays  suffer  a  deflection  d  in  travelling  a  distance  s 

2  dv* 

through  this  field,  /=  -— —  > 

s 

and  the  ratio  of  charge  to  mass  is  given  in  terms  of  measurable 
quantities  by  the  expression 


m-E?- 

Similarly,  if  the  electric  field  be  suppressed,  the  value  of  the 
ratio  of  charge  to  mass  may  be  obtained  from  the  deflection 
suffered  in  traversing  a  magnetic  field.  In  this  case 

e       2dvc 


By  these  methods  it  is  found  that 
for  the  negative  electron. 


-  =  1.77(10)7<?V4^  (37) 


58        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Determinations  of  this  ratio  for  beta  rays  moving  with  veloci- 
ties only  slightly  less  than  the  velocity  of  light  verify  the  theo- 
retical expression  (20)  for  the  increase  of  transverse  mass  with 
velocity. 

In  order  to  measure  e  and  hence  m  other  experimental 
methods  are  necessary.  Suppose  an  electron  to  be  attached  to 
a  minute  oil  drop  situated  between  the  horizontal  plates  of  a  par- 
allel plate  condenser.  If  the  electric  field  is  adjusted  so  that  the 
oil  drop  remains  at  rest,  its  weight  w  is  balanced  by  the  force 
eE.  Now  if  the  drop  is  allowed  to  fall  freely  through  the  sur- 
rounding gas,  its  radius  may  be  calculated  from  its  rate  of  fall 
by  Stokes'  law.  From  the  radius  and  density  of  the  oil  w  may 
be  determined  and  hence  e  computed.  In  this  way  the  electronic 
charge  has  been  found  by  Millikan  to  be 

*  =  4.77(10)-10V4^,  (38) 

and,  combining  this  with  (37),  the  mass  of  the  negative  electron 

is  found  to  be  n  A  ..,  A,    2s  ^on\ 

m  =  9.0  (10)~28  gm.,  (39) 

which  is  about  one  eighteen-hundredth  of  the  mass  of  a  hydro- 
gen atom.    Hence  the  radius  of  the  negative  electron  is 

a  =  1.88  (10)- 13  cm. 

Since  the  positive  electron  has  not  yet  been  isolated,  it  has 
been  impossible  to  measure  its  mass  and  radius,  although  there 
are  reasons  for  supposing  that  it  has  a  much  greater  mass  and 
consequently  a  much  smaller  radius  than  the  negative  electron. 


CHAPTER  V 

EQUATIONS  OF  THE  ELECTROMAGNETIC  FIELD 

24.  Divergence  equations.  The  field  due  to  a  point  charge  is 
completely  specified  by  any  two  of  the  three  vector  functions  of 
position  in  space  and  time,  E,  H,  and  c.  In  the  case  of  a  com- 
plex field,  E  and  H  are  the  resultants  of  the  electric  and  mag- 
netic intensities  respectively  of  the  component  simple  fields,  but 
c  must  be  given  for  each  elementary  field.  Hence  in  order  to 
avoid  explicit  reference  to  the  components  of  a  complex  field, 
as  well  as  in  order  to  give  the  field  equations  as  great  a  sym- 
metry as  possible,  the  field  is  usually  described  in  terms  of 
E  and  H. 

To  find  the  divergence  of  the  electric  intensity  due  to  any 
given  distribution  of  point  charges  consider  a  small  region  dr 
surrounded  by  the  closed  surface  <r.  By  Gauss'  theorem, 

V-Edr  = 

Let  letters  with  strokes  over  them  refer  to  component  fields. 

m  dN  , 

Then  Eda  =  —fa 

eta 

for  each  element  of  charge. 

Therefore  V-Ec?r=V  C  dN. 


Now  the  part  of  this  sum  due  to  charges  outside  the  region 
dr  vanishes,  while  the  part  due  to  the  charge  de  inside  this 
region  becomes  equal  to  de  itself.  Hence 


(1) 


where  p  is  the  density  of  charge  at  the  point  in  question. 

59 


60         AN  INTRODUCTION  TO  ELECTRODYNAMICS 

In  section  13  it  was  shown  that 

H=Vxa. 
Therefore  V-H  =  0  (2) 

identically,  or  the  divergence  of  the  magnetic  intensity  is  every- 
where zero. 

25.  Vector  fields.  Any  vector  function  of  position  in  space 
and  time  may  be  represented  by  moving  lines  such  as  have 
been  employed  to  give  a  geometrical  significance  to  the  electric 
intensity.  These  lines  will  be  continuous  at  all  points  where 
the  divergence  of  the  vector  function  vanishes,  as  is  obvious 
from  the  discussion  contained  in  the  preceding  section.  At 
other  points  lines  will  either  begin  or  end.  Equation  (1)  shows 
that  the  resultant  electric  intensity  may  be  represented  by  con- 
tinuous lines  at  all  points  except  those  at  which  electricity  is 
present,  while  equation  (2)  shows  that  the  resultant  magnetic 
intensity  may  be  represented  everywhere  by  continuous  lines. 

Let  V  be  a  vector  function  whose  magnitude  and  direction 
are  represented  by  lines  all  points  of  which  are  moving  with 
velocities  of  the  same  magnitude  c.  If  dN  tubes  of  these  lines 
(a  tube  being  a  bundle  of  M  lines)  pass  through  a  small  surface 
dsx  with  normal  parallel  to  the  X  axis, 


For  the  moment  assume  that  no  new  lines  are  formed.  Then 
in  a  time  dt,  Vx  may  suffer  a  change  due  to  three  causes.  In 
the  first  place,  the  number  of  lines  passing  through  dsx  may 
increase  by  virtue  of  the  fact  that  the  lines  whose  motion  will 
bring  them  through  this  surface  at  the  end  of  the  time  dt  are 
more  closely  packed  than  those  passing  through  dsx  originally. 
The  increase  in  F  due  to  this  cause  is 


Secondly,  the  velocity  associated  with  the  new  lines  may  have 
a  different  direction  from  that  of  the  old.    This  will  produce  a 


EQUATIONS  OF  ELECTKOMAGNETIC  FIELD         61 


crowding  of  the   lines  during  the  time  dt  and   account  for  a 
change  in  Vx  equal  to 


Finally,  if  c  differs  in  direction  at  neighboring  points  on  the 
same  line,  there  will  ensue  a  twisting  of  the  lines  which  will 
produce  in  Vx  an  increase 


in  the  time  dt. 

Therefore  the  total  rate  of  change  of  Vx  is 


JL*  =  _  c.  V  V  - 


+  V- 


or 


JL  +  cV-V=Vx{cxV}. 

ot 


(3) 


Now  consider  the  increase  in  V  due  to  the  formation  of 
new  lines.  Attention  will  be  confined  to  those  fields  whose 
lines  terminate  only  on  sources.  Let 
the  points  O  etc.  in  Fig.  9  be  each  the 
source  of  a  new  line  emitted  in  the  direc- 
tion of  the  arrows,  the  line  sources  them- 
selves having  a  velocity  v  to  the  right. 
The  number  of  tube  sources  per  unit 
volume  is  obviously 


FIG.  9 


and  as  the  direction  of  the  lines  at  P 
is  QP,  the  increase  in  the  value  of 
V  at  P  in  a  time  dt  due  to  the  formation  of  new  lines  is 


Therefore  the  complete  expression  for  the  rate  of  change  of 
V  becomes 

~  +  vV-V=Vx{cxV}.  (4) 

ct 


62        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

If  the  field  under  consideration  is  due  to  the  superposition 
of  a  number  of  simple  fields  of  the  character  of  those  just  dis- 
cussed, 

(5) 


where  the  stroked  letters  refer  to  the  component  fields.  More- 
over, since  a  number  of  point  sources  emitting  lines  in  different 
directions  may  be  considered  to  coalesce,  this  equation  applies 
equally  well  to  point  sources  from  which  lines  diverge  in  all 
directions.  It  is  to  be  noted  that  equation  (5)  is  a  consequence 
of  the  properties  of  three-dimensional  space,  and  nothing  more. 
26.  Curl  equations.  Replace  V  by  E  in  (5).  Substituting 
from  (1),  the  second  term  of  the  left-hand  member  becomes 


.....  . 

Since,  however,  only  one  element  of  charge  may  occupy  one 
point  in  space  at  a  given  time,  the  summation  sign  may  be 
dropped. 

The  right-hand  side  of  (5)  becomes 


where  CE  is  the  velocity  of  a  moving  element  of  an  electric  line 
of  force  of  one  of  the  component  fields. 


But  HEE  - 

Hence  VxH  =  -{E  +  pv>.  (6) 

If  V  is  replaced  by  H  in  (5),  it  follows  from  (2)  that  the 
second  term  of  the  left-hand  member  vanishes.  Hence  the 
equation  becomes  -i  1 

VxS£fe.xfi}-J*,  (7) 

where  the  velocity  c^  of  a  moving  element  of  a  magnetic  line 
of  force  of  one  of  the  component  fields  does  not  in  general  have 
the  same  direction  as  the  velocity  CE  of  a  moving  element  of  an 
electric  line  of  force  of  the  same  field. 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         63 

But  if  a  is  eliminated  from  the  equations  obtained  by  curling 
(20)  and  differentiating  (21)  with  respect  to  the  time,  in  section  13, 
it  is  found  that  i  . 

Hence,  as  (7)  and  (8)  apply  equally  well  to  simple  or  com- 
plex fields,  comparison  of  these  two  equations  shows  that  for  a 
single  elementary  field  i 

i-i  •*-  ^       .  ..   -rr     *      **    i  /Q\ 

w 


c 
a  relation  which  is  complementary  to  the  definition 

I=-e£xE.  (10) 

c 
_  1  _          

Therefore  E  =  -  -5  %  x  (c£  x  E)  +  Vi/r, 

or 


and 

or  H(l- 


Hence  it  follows  that  CH  is  parallel  to  CE  when,  and  only 
when,  V^  is  parallel  to  CE. 

27.  Electrodynamic  equations.  Equations  (1),  (2),  (6),  (8) 
specify  the  electromagnetic  field  in  terms  of  the  distribution 
of  density  of  charge  p  and  velocity  v.  If  (2),  section  18,  is 
written  as  an  equation  instead  of  as  a  definition  in  order  to 
signify  that  it  includes  the  dynamical  assumption  immediately 
following  it,  this  equation  suffices  to  determine  the  effect  of 
an  electromagnetic  field  upon  matter.  These  five  equations 
contain  the  whole  of  electrodynamics.  Collected  they  are: 

V-E  =  p,  (11)         V-H  =  0,  (13) 

VxE=-in,  (12)     VxH  =  -{E  +  /3v},  (14) 


(15) 
where  F  is  the  electromagnetic  force  per  unit  volume. 


64        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
28.  Energy  relations.    From  (12)  and  (14)  it  follows  that 

H-V  xE-E-VxH  =  --  (E-E  +  H-H)-  -  oE-v. 

c  o 

But  H-V  x  E  -  E-V  x  H  =  V-(E  x  H), 

and  />E«v  =  F«v. 

Hence     %-  {\  (E*  +  #2))  +  <?V-(E  x  H)  +  F-v  =  0. 


Integrating  over  any  arbitrarily  chosen  portion  of  space,  and 
applying  Gauss'  theorem  to  the  second  term, 


f  (E  x  H)-<for  +  f  F-vdr  =  0,    (16) 


where  dr  is  an  element  of  volume,  and  da-  an  element  of  the 
bounding  surface  having  the  direction  of  the  outward  drawn 
normal.  The  third  term  of  this  expression  measures  the  rate  at 
which  work  is  done  %  the  electromagnetic  field  on  the  matter 
contained  in  the  region  selected.  Hence,  if  the  principle  of  con- 
servation of  energy  is  to  hold,  the  first  two  terms  must  be 
interpreted  as  the  rate  at  which  the  energy  of  the  field  increases 
plus  the  rate  at  which  energy  escapes  through  the  surface 
bounding  the  field. 

Suppose  E  and  H  to  be  zero  everywhere  on  this  surface. 
Then  no  energy  escapes  from  the  region  enclosed,  and  the  rate 
at  which  work  is  done  on  the  electromagnetic  field  equals  the 
rate  at  which  its  energy  increases.  But  the  integrand  of  the 
second  term  of  (16)  is  everywhere  zero.  Hence  the  rate  of 
increase  of  energy  of  the  field  must  be  represented  by  the  first 
term,  the  form  of  which  suggests  that 

*»}(**+£*) 
is  to  be  considered  as  the  electromagnetic  energy  per  unit  volume. 

The  second  term  of  (16),  then,  must  be  interpreted  as  the  out- 
ward flux  of  energy  through  the  surface  enveloping  the  field,  and 
its  form  suggests  that  s  =  c(Ex'B) 

is  to  be  considered  as  the  flow  of  energy  per  unit  cross  section 
per  unit  time. 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         65 

29.  Electromagnetic  waves  in  space.    For   empty  space  the 
electromagnetic  equations  take  the  form 

V-E  =  0,  (17)  V-H  =  0,  (19) 

VxE  =  --H,       (18)  VxH  =  iE.  (20) 

0  C 

To  eliminate  H,  curl  (18)  and  differentiate  (20)  with  respect 
to  the  time.    Thus 

V  x  V  xE  =  --  V  x  H 

c 


But  V  X  V  x  E  =  VV-E  -  V-VE 

=  _  V-VE. 

Therefore  V-VE  -  \  E  =  0,  (21) 

(T 

and,  similarly,  V-  VH  -  1  H  =  0.  (22) 

These  are  equations  of  waves  moving  with  velocity  c. 
Consider  a  plane  wave  advancing  along  the  Xaxis.    Then  E 
is  a  function  of  x  and  t  only,  and  (21)  reduces  to 

The  solution  of  this  equation  for  the  case  of  a  wave  moving 
in  the  positive  X  direction  is 


Hence  it  follows  from  (17)  that  Ex  is  a  constant,  and  since 
the  present  discussion  is  concerned  only  with  the  variable  part 
of  the  field,  this  constant  may  be  taken  as  zero. 


66         AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Therefore  Ex  =  0,  (24) 

Ey=g(x-ct\  (25) 

EK=h(x-ct).  (26) 

From  (18)  it  follows  that 


and  hence  Hx=  0,  (27) 

Hy  =  -h(x-cfy  (28) 

Hn=g(x-c£),  (29) 

except  for  a  possible  constant  of  integration.  Therefore  the 
variable  parts  of  E  and  j£T  have  the  same  magnitude  at  any 
point  and  time,  and  lie  in  a  plane  at  right  angles  to  the  direc- 
tion of  propagation. 

The  cosine  of  the  angle  between  E  and  H  is  proportional  to 


=  0, 

showing  that  E  and  H  are  at  right  angles  in  the  wave  front. 
The  preceding  section  gave  the  flow  of  energy  as 


(30) 

showing  that  the  propagation  of  energy  is  along  the  X  axis  and 
that  the  entire  energy  of  the  wave  front  is  advancing  with 
the  velocity  of  light.  It  follows  from  this  equation  that  the 
direction  of  propagation  of  the  wave  is  at  right  angles  to  the 
plane  of  E  and  H  in  the  sense  of 

ExH, 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         67 

Consider  a  spherical  wave  having  its  center  at  the  origin.  In 
this  case  E  is  a  function  of  the  radius  vector  r  and  t  only, 
and  (21)  reduces  to 

?2E      2?E      1PE      „  „ 

g?  +  r?;~?a?=  °' 

of  which  the  solution  is 


As  in  the  case  of  the  plane  wave,  it  may  be  shown  that 
E  and  H  are  mutually  perpendicular,  and  at  right  angles  to  the 
direction  of  propagation.  If  r±  is  a  unit  vector  along  the  out- 
ward drawn  normal  to  the  wave  front,  the  flow  of  energy  is 


(32) 

showing  that  the  amount  of  energy  passing  through  unit  cross 
section  in  unit  time  varies  inversely  with  the  square  of  the  dis- 
tance from  the  source,  and  that  the  entire  energy  of  the  wave 
front  is  advancing  with  the  velocity  of  light. 

30.  Radiation  pressure.  Substituting  in  (15)  the  values  of  p 
and  pv  from  (11)  and  (14),  the  electromagnetic  force  per  unit 
volume  takes  the  form 

F  =  V-EE  +  (V  X  H)  x  H  -  i  E  X  H, 

c 

and  making  use  of  (12)  and  (13), 

F  =  -1^(E 

c  dt 


68        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

The  total  electromagnetic  force  on  the  matter  in  any  given 
region  is  „ 

*-/' 


where  K,  =--  C^(E  x  H)dr 

cj  at 

— ?!/"•*••  <33> 

and  Ka=f{VvEE+V-HH  +  (VxE 

Now  let  J  =  f  {V-EE+(VxE)xE}d 
=  f  (V-EE  +  E-VE  -  i 

=  f{V-(EE)-lV( 

By  Gauss'  theorem, 

fv-(EE)^r=  TEE-^O-, 

and  f  V  (E-E)  dr  =  i  C V-  (iE-E)  dr  -f  etc. 

=  i  f  E-Ei-c?o-  +  etc. 

=  TE-E^O-. 

Therefore     J  =  CEE-dv  -  %  fE-Ecfo-, 
and  Kv  is  given  entirely  by  the  surface  integral 

Ka  =  f(EE  +  HH>dcr  -  1  f(E-E  +  H-H)  dtr      (34) 

taken  over  the  surface  bounding  the  chosen  region. 

Consider  a  region  which  is  the  seat  of  either  stationary  radia- 
tion or  undamped  periodic  vibrations.    In  the  former  case  the 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         69 

average  KT  is  zero,  and  in  the  latter  case  KT  vanishes  provided 
the  value  of  the  force  desired  is  the  average  over  a  whole  num- 
ber of  periods.  Hence  the  force  on  the  matter  within  the  region 
is  given  entirely  by  K^. 

Consider  a  box  (Fig.  10)  with  perfectly  reflecting  walls  con- 
taining homogeneous  isotropic  radiation.  Describe  the  pill-box 
shaped  surface  ABCD  about  an  element  dcr  of  that  part  of  the 
wall  of  the  box  which  is  perpendicular  to  the  X  axis.  The  only 


o 


-A' 


FIG.  10 

matter  within  this  closed  surface  is  the  portion  MN  of  the  wall 
of  the  box.  In  so  far  as  the  effect  of  the  radiation  inside  the 
box  is  concerned,  the  average  force  on  MN  is  given  entirely  by 
Ka,  which  reduces  to  an  integral  over  BC,  since  AD  lies  com- 
pletely outside  the  field,  and  the  sides  AB,  CD  of  the  pill-box 
have  a  negligible  area.  Hence  if  Xx,  Fa;,  Zx  are  the  forces  per 
unit  area  on  the  surface  MN  of  the  box  due  to  the  pressure  of 
the  radiation  inside  it,  parallel  respectively  to  the  X,  F,  Z  axes, 

^-H^,  (35) 
(36) 
(37) 

and  similar  expressions  hold  for  the  parts  of  the  wall  perpen- 
dicular to  the  Y  and  Z  axes. 


70        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
As  the  radiation  is  assumed  homogeneous  and  isotropic, 


where  the  stroke  denotes  the  average  value  of  the  quantity  over 
which  it  is  placed.  Similar  relations  hold  for  the  components 
of  the  magnetic  intensity. 
Therefore 


=  -iu,  (38) 

YX=ZX  =  0,  (39) 
showing  that  the  radiation 
within  the  box  exerts  a 
normal  stress  on  the  walls 
equal  to  one  third  of  the 
electromagnetic  energy  per 
un'it  volume. 

Consider,  now,  a  box 
similar  in  shape  to  that  of 
Fig.  10  with  perfectly  con- 
ducting walls.  Instead  of  being  filled  with  homogeneous  radia- 
tion, suppose  the  box  to  contain  a  train  of  undamped  plane 
waves  which  are  incident  on  the  surface  MN  (Fig.  11)  at  the 
angle  6. 

Referred  to  the  ^Y^  axes  in  the  figure,  equations  (24)  to 
(29)  give  for  the  incident  wave 


FIG.  11 


and 


H  =  0, 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         71 

Therefore    the    components    of    the    electric    and    magnetic 
intensities    along   the    XYZ  axes   due    to    the    incident   wave 

akme'    are 


Hx=-  \sa\0, 


where  the  argument  of  the  arbitrary  gl  and  ^  functions  is 

xl  —  ct  =  —  x  cos  9  —  y  sin  6  —  ct. 

Similarly,  for  the  reflected  wave, 


where  the  argument  of  the  g2  and  h2  functions  is 

x2  —  ct  =  x  cos  6  —  y  sin  6  —  ct. 

At  the  reflecting  surface  the  arguments  of  the  functions  with 
different  subscripts  become  the  same.  Moreover,  as  this  surface 
is  perfectly  conducting,  the  tangential  component  of  the  resultant 
electric  intensity  must  vanish.  Therefore 

=  0, 


Hence  it  follows  from  (30)  that  all  the  energy  brought  up 
in  the  incident  wave  is  carried  away  in  the  reflected  wave. 


72    AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Therefore  a  perfectly  conducting  surface  is  a  perfect  reflector  of 
electromagnetic  waves. 

The  resultant  electric  and  magnetic  intensities  just  to  the 
right  of  MN  are  ^=  %g  sinft  (40) 

tf,=  0,  (41) 

J?.=  0,  (42) 

^,=  0,  (43) 

^y=27icos0,  (44) 

#,=  2tf.  (45) 

Hence,  as  there  is  no  field  to  the  left  of  MN,  this  surface 
must  have  a  charge  2  g  sin  0  per  unit  area.  If  g  is  a  simple 
harmonic  function,  this  charge  will  be  alternately  positive  and 
negative. 

Substituting  the  values  (40)  to  (45)  in  the  expressions  (35), 
(36),  (37)  for  the  components  of  the  stress  on  the  reflecting 

SUrf ace'  Xx  =  -  2  (/  +  A2)  cos2  0,  (46) 

*-,  =  *,=  0,  (47) 

showing  that  the  stress  due  to  the  radiation  is  a  pressure  nor- 
mal to  the  surface. 

Now,  the  energy  density  of  the  incident  wave  is 

and  therefore  Xx  =  -  2  ^  cos2  0.  (48) 

Let  /  stand  for  the  energy  striking  the  reflecting  surface  per 
unit  area  per  unit  time.  Then 

I=u^c  cos  0, 
and  Xx  =  -  2  -  cos  0.  (49) 

Suppose  that  instead  of  containing  a  single  train  of  plane 
waves,  the  box  is  filled  with  plane  waves  travelling  in  all  direc- 
tions at  random  ;  that  is,  with  a  homogeneous  isotropic  radiation. 
Then  if  u  is  the  energy  density  of  the  radiation,  the  energy  per 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         73 


unit  volume  of  that  portion  of  the  radiation  which  is  incident 
on  MN  between  the  angles  0  and  6  +  dO  is 


^  ==  -  u  sin  6d6, 


and 


— 


cos2  6  sin  6d6 


agreeing  with  (38). 

31.  Electromagnetic  momentum.  Consider  a  closed  surface 
A  BCD  (Fig.  12),  surrounding  some  matter.  Let  the  field  inside 
this  surface  be  in  a  stationary  state.  Then  the  force  on  the  matter 
contained  is  given  by  K^  integrated  over  the  bounding  surface, 
the  outward-drawn 
normal  being  posi- 
tive. Denote  by  A  M 
Kai  the  part  of  Kff 
obtained  by  inte- 
grating over  BC. 
Then  the  force  K 
on  the  matter  under 
consideration,  due 
to  the  electromag- 
netic field  extending 
through  the  surface  FlG  12 

J5(7,  is  equal  to  K^. 

Let  BCEF  be  a  closed  surface  which  surrounds  no  matter, 
such  that  the  radiation  field  inside  it  comes  into  contact  with  the 
surface  only  between  B  and  C.  Then  the  value  of  K^  integrated 
over  this  surface  is  equal  to  —  K^,  since  the  outward-drawn 
normal  to  BC  in  this  case  has  the  opposite  direction  to  that  for 


D 


N 


ABCD.    Hence 


-K, 


as  no  matter  is  present  inside  the  surface.    Therefore 

K-KT=0, 


(50) 


74         AN  INTRODUCTION  TO  ELECTRODYNAMICS 

where  K  is  the  force  exerted  by  the  electromagnetic  field  in 
BCEF  on  the  matter  in  ABCD.  Hence,  if  the  law  of  action  and 
reaction  is  to  hold,  —  KT  must  be  interpreted  as  the  force  exerted 
by  the  matter  in  AS  CD  on  the  electromagnetic  field  in  BCEF. 
The  form  of  this  expression,  namely, 


-*,=!{!  f 

r      dt  \c2J 


suggests  that  -^s 

is  to  be  considered  as  the  momentum  per  unit  volume  of  the 
electromagnetic  field. 

Suppose  the  matter  in  ABCD  to  consist  of  a  rigid  body  with 
a  plane,  perfectly  reflecting  surface  just  to  the  left  of  BC.  Let 
a  limited  train  of  plane  waves  be  incident  on  this  surface  at 
the  angle  6.  Reference  to  (30)  shows  that  the  electromagnetic 
momentum  per  unit  volume  of  the  incident  radiation  is 


c2 

Therefore  the  momentum  of  the  radiation  striking  each  unit 
area  of  the  surface  in  unit  time  is  u^  cos  0,  and  the  momentum 
of  the  reflected  radiation  is  of  the  same  magnitude.  The  force 
exerted  by  each  unit  area  of  the  reflecting  surface  on  the  train 
of  waves  is  equal  to  the  vector  increase  in  momentum  per  unit 
time  ;  that  is,  -  K  =  2u  cos2 6  • 

T  1 

along    the    outward-drawn    normal.     Consequently    the    stress 
exerted  by  the  radiation  on  this  surface  is 

=  -2-cos<9, 

c 
agreeing  with  (48)  and  (49). 

32.  Four-dimensional  representation.  In  four-dimensional  space 
two  mutually  perpendicular  lines  may  be  drawn  at  right  angles 
to  an  element  of  surface.  Consequently  the  vector  properties 
of  such  an  element  cannot  be  expressed  by  the  direction  of  its 
normal.  In  fact,  it  is  necessary  to  distinguish  between  directed 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         75 

linear  elements,  directed  surface  elements,  and  directed  volume 
elements.  The  first  have  as  components  their  projections  on 
the  four  coordinate  axes,  the  second  their  projections  on  the 
six  coordinate  planes,  and  the  third  their  projections  on  the  four 
coordinate  planoids.  Hence  the  first  and  third  are  often  called 
four-vectors,  and  the  second  six  -vectors.  Here  they  will  be  called 
vectors  of  the  first,  second,  and  third  orders  respectively. 

Let  a,  b,  and  c  be  three  vectors  of  the  first  order.  Then  the 
vector  product  a  x  b  is  a  directed  surface  having  for  its  area  the 
parallelogram  of  which  a  and  b  are  the  sides,  and  so  directed  that 

bxa  =  —  axb. 

Similarly,  a  X  b  x  c  is  a  directed  volume  having  for  its  magni- 
tude that  of  the  parallelepiped  whose  edges  are  a,  b,  and  c.  In 
the  case  of  any  cross  product,  interchange  of  two  adjacent  vectors 
changes  the  sign  of  the  product. 

Let  kt,  k2,  k3,  k4  be  unit  vectors  of  the  first  order  parallel 
respectively  to  the  X,  J,  Z,  L  axes.  Then 

*»sk.x*. 

is  a  unit  vector  of  the  second  order  in  the  YZ  coordinate  plane,  and 


is  a  unit  vector  of  the  third  order  in  the  ZLX  coordinate 
planoid.  From  their  definition  it  is  obvious  that  two  adjacent 
subscripts  of  a  unit  vector  of  the  second  or  third  order  may  be 
interchanged  provided  the  sign  of  the  vector  is  changed. 

The  dot  product  of  two  unit  vectors  is  defined  as  follows. 
If  the  vector  of  lower  order  has  a  subscript  which  the  other 
lacks,  the  dot  product  vanishes.  Otherwise,  the  product  is  the 
unit  vector  remaining  after  like  digits  in  the  subscript  of  each 
vector  have  been  brought  to  the  end  and  cancelled.  Thus 


- 


76        AN  INTKODUCTION  TO  ELECTRODYNAMICS 

Thus,  while  the  order  of  the  cross  product  of  two  unit  vectors 
is  equal  to  the  sum  of  their  orders,  the  order  of  the  dot  product 
is  equal  to  the  difference  of  the  orders  of  the  two  factors. 

Consider  the  vector  of  the  second  order 


.    (52) 

The  dual  Mx  of  M  is  defined  as  a  vector  of  the  same  order 
with  components  such  that 

ML  =  MV, 

where  mnop  is  formed  from  xyzl  by  an  even  number  of  inter- 
changes of  adjacent  letters.    Hence 

M*  ==  Jf2,k12  +  MjLn  +  My^  +  MJ^  +  MJt^  +  MJ^.  (53) 
If  P  =  ^  +  J?kt+^k8-+JJk4,  (54) 

the  rule  for  forming  the  dot  product  shows  that 
P-M  s 


-P,Mzl  )k4.  (55) 

In  four-dimensional  analysis  the  vector  operator 


plays  much  the  same  part  as  V  in  three  dimensions. 
Consider  the  product 


cy          cz          cl 

+  ~af  ~ 


dx         cy  c 

2* 2? 2?  )k.  (56) 

cx         dv          dz  ] 


EQUATIONS  OF  ELECTROMAGNETIC  FIELD         77 

Remembering  that  I  =  ict,  equations  (11)  and  (14)  of  the 
electromagnetic  field  may  be  written 

*  cH       dH       diE       1 

i  -  ?  --  v.  --  x.  =  __  QV 

dy        dz         dl        cH 
dH  dH       diE       1 


dx  dz         Ol        c 


_ 

dx  "~dy 

diEx      BiK       diEz 

^  +  "a"^  +  -^ 

cx         dy         dz 
and  equations  (12)  and  (13) 


diEx  _0 

dx  dz    "   dl 


*  =  Q 
dy  ~  m'~ 


_          __ 

dx         dy         dz 

Comparison  with  (52),  (53),  (54),  and  (56)  shows  that  if 


and       P  =  -   v      +  - 


O 


the  two  scalar  equations  (11)  and  (13)  and  the  two  vector  equa- 
tions (12)  and  (14)  of  the  electromagnetic  field  are  expressed 
by  the  pair  of  four-dimensional  vector  equations 

0'M  =  P,  (57) 

0'M*=0.  (58). 


78        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

The  equation  (15),  which  gives  the  effect  of  an  electro- 
magnetic field  on  matter,  and  the  energy  equation  may  be  written 
together  in  the  form  9 

Fx  =  +  -  PvyHg  -  -  PvzHy  +  PEX1 

G  6 


Fz  =      ~  pvxHy  -  -  pvyffx 

G  6 

ijt  =  \ 

Hence,  if         F= 

C 

these    relations    are    contained   in  the   four-dimensional  vector 
equation  F  =  p.M_  (59) 

The  scalar  operator 


is  known  as  the  d'Alembertian.    In  four-dimensional  analysis 
the  wave  equations  (21)  and  (22)  are  expressed  by 

$-OM  =  0.  (60) 

In  addition  to  those  just  mentioned,  many  other  electrody- 
namic  relations  may  be  expressed  in  far  more  compact  form  in 
terms  of  four-dimensional  vector  analysis  than  in  the  analysis  of 
three  dimensions. 


CHAPTER  VI 

RADIATION 

33.  Radiation  from  a  single  electron.    Let  v  and  f  be  the 

electron's  velocity  and  acceleration  respectively  at  the  time  t. 
Take  its  position  at  this  time  as  origin,  and  choose  axes  so  that 
the  X  axis  is  parallel  to  v  and  the  XY  plane  contains  f.  Describe 
about  the  origin  a  sphere  of  radius  r,  large  compared  with  the 
radius  of  the  electron.  Consider  an  element  da-  of  the  surface 
of  this  sphere  such  that  the  radius  vector  drawn  to  it  from  the 
origin  makes  an  angle  6  with  the  X  axis.  The  energy  emitted 
from  the  electron  during  an  interval  of  time  dt  in  the  direction 
of  this  radius  vector  will  reach  the  surface  of  the  sphere  at  a 
time  t  +  r/c,  and  will  take  a  time 


to  pass  through  this  surface.    Hence,  as  the  flow  of  energy  per 
unit  cross  section  per  unit  time  is  equal  to 


the  energy  emitted  by  the  electron  during  the  time  dt  is  given 

by  the  integral  /* 

Rdt=  [  s- 

taken  over  the  surface  of  the  sphere.    Therefore  the  rate  of 
radiation  from  the  electron  is 

R=  fs.<fcr(l-/3cos0).  (1) 

As  H  =  -  c  x  E, 

c 

it  follows  that  s  =  E\  -  E-cE. 

The  sphere  over  which  the  surface  integral  is  to  be  taken 
may  be  made  as  large  as  desired.    Consequently  the  terms  in 

79 


80        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

equation  (14),  section  12,  for  the  electric  intensity  which  involve 
the  inverse  square  of  the  radius  vector  may  be  made  so  small 
compared  with  the  term  involving  the  inverse  first  power  of 
this  quantity  that  they  may  be  neglected.  As  the  term  involv- 
ing the  inverse  first  power  defines  a  vector  perpendicular  to  c, 


f  f  ,  2/yj.  (1-W,2      1         (2) 

l(l-/3cos604      (l-/3cos<9)5      (l-/3cos<9)6J 

where  fr  is  the  component  of  the  electron's  acceleration  along  the 
radius  vector.   For  small  values  of  0  this  becomes  approximately 


_ 
= 


16  -rrW 


where  a  is  the  angle  which  the  radius  vector  makes  with  f. 
Hence  the  radiation  vanishes  in  the  direction  of  f  and  is  maxi- 
mum at  right  angles  to  this  direction. 

To  find  the  rate  of  total  radiation  from  the  electron,  integral 
(1)  must  be  evaluated.    Substituting  the  expression  for  s  given 

by  (2), 

C       da         \»B  C     f' 

J  r2(l-/3cos<?)3      'PJ  r>(l- 


»= 


f 
\ 


(i  -  /s2 

If  f  is  the  acceleration  of  the  electron  at  the  time  t  relative 
to  the  system  in  which  it  is,  at  that  instant,  momentarily  at 
rest,  reference  to  equations  (27),  (28),  (29),  section  6,  shows  that 


34.  Radiation  from  a  group  of  electrons.  In  the  calculations 
of  this  section  it  will  be  assumed  that  the  greatest  distance 
between  two  electrons  of  the  group  is  small  compared  with  the 
wave  length  of  the  radiation  emitted,  and  that  the  velocity  of 
the  fastest  moving  electron  of  the  group  is  small  compared  with 


RADIATION  81 

that  of  light.  If  a  sphere  of  radius  r,  large  compared  with  the 
dimensions  of  the  group,  is  described  about  some  point  in  the 
group  as  center,  the  energy  passing  through  the  surface  of  this 
sphere  in  unit  time  is  found  as  in  the  last  article  to  be 


(5) 

J 

where  s  = 


v 

the  summation  being  taken  over  the  n  electrons  in  the  group. 


AT  TT      e{fi»cc  —  c2fi} 

Now  j?  =_L^ — »  (6) 

4  7m?4 

at  a  great  distance  from  the  ith  electron. 

Therefore       s  =  1  /,  a  ^{<?fffj- f^cfyc}.  (7) 

Suppose  that  the  sum  of  the  components  of  the  accelerations 
in  any  direction  is  equal  to  that  in  the  opposite  direction. 
Then  s  vanishes  for  all  directions  of  c.  Hence  a  ring  of  any 
number  of  evenly  spaced  electrons  which  are  rotating  about  a 
common  axis  with  constant  speed,  will  emit  no  appreciable 
radiation. 

To  get  the  total  radiation  from  the  group  of  electrons,  sub- 
stitute (7)  in  (5)  and  integrate.  In  this  way  it  is  found  that 


35.  Energy  of  a  moving  electron.    The  dynamical  equation  of 
an  electron  has  been  shown  to  be 

K  =  mi  -  ni, 
through  terms  of  the  third  order,  where 


82        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

The  work  done  on  the  electron  in  a  time  t  by  the  impressed 
field  is 

W=    CK-vdt 

=  m  Cfadt  -  n  Cf*vdt 

=   ™  0*  -  *D  -  "(v2.f2  -  Vl.f  j  +         >*       (9) 


where  the  subscript  1  denotes  the  value  of  the  quantity  to 
which  it  is  attached  at  the  beginning  of  the  time  t  and  the  sub- 
script 2  the  value  of  this  quantity  at  the  end  of  this  time. 
The  first  term  in  this  expression  represents  the  kinetic  energy 
of  the  electron,  the  second  term  its  acceleration  energy,  and  the 
third  term  the  energy  which  has  been  radiated.  The  first  two 
terms  are  reversible  in  the  sense  that  the  energy  which  they 
represent  may  be  recovered  when  the  electron  returns  to  its 
original  state  of  motion,  but  the  third  is  irreversible. 

Consider  an  electron  which  starts  from  rest  and  acquires  a 
velocity  v  by  virtue  of  a  very  small  acceleration  continued  for 
a  very  long  time.  By  making  the  derivatives  of  the  accelera- 
tion small  enough,  the  second  and  succeeding  terms  of  the 
dynamical  equation  may  be  made  as  small  as  desired  com- 
pared with  the  first  term.  Hence,  if  account  is  taken  of  the 
variation  of  mass  with  velocity,  this  equation  is  given  to  any 
desired  degree  of  accuracy  by 


and  the  work  done  is  expressed  as  closely  as  may  be  desired  by 
W=  f  K-v<ft 


(10) 


RADIATION  83 

In  this  case  the  energy  radiated  is  inappreciable,  so  all 
the  energy  acquired  by  the  electron  may  be  recovered  if 
it  is  brought  back  to  its  original  state  of  rest  by  a  similar 
process. 

Now  the  energy  of  an  electron  moving  with  constant  velocity 
v  is  given  by 

Inside  the  electron's  surface  both  E  and  H  vanish,  while  out- 
side the  surface  the  values  of  these  intensities  are  given  by 
equations  (3)  and  (4),  section  19.  Hence  if  the  angle  which  the 
radius  vector  makes  with  the  direction  of  the  electron's  velocity 
is  denoted  by  0, 

_e\\-py 

~[^r 

where  r  goes  from 


Vl-/32sin20 
to  infinity,  and  0  from  0  to  TT. 


Integrating,  ^==-1,  (11) 

and  the  increase  in  energy  is  given  by 


The  discrepancy  between  equations  (10)  and  (12)  arises  from 
the  fact  that  in  the  calculation  from  which  the  former  was 
obtained  no  account  was  taken  of  the  work  done  against  the 
electron's  field  in  connection  with  the  progressive  contraction 
which  takes  place  as  its  velocity  increases.  In  order  to  deter- 
mine the  work  done  in  this  process  of  contraction,  it  is  neces- 
sary to  evaluate  the  stress  K  on  each  unit  area  of  the  electron's 
surface.  From  the  expressions  for  the  electric  and  magnetic 


84        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

intensities  given  in  section  19  it  follows  that  the  electromag- 
netic force  per  unit  volume  just  outside  the  surface  is  given  by 


pe      (l-/32)cos(? 


pe     (l-/32)2sin<9 


where  the  X  axis  is  taken  in  the 
direction  of  the  electron's  velocity. 
If  a  (Fig.  13)  is  the  angle  which 
the  normal  to  the  surface  makes 
with  the  X  axis, 

tan  a  =  (1  —  /32)  tan  6.  FIG.  13 

Therefore  the  tangential  component  Ft  of  F  is  given  by 
Ft  =  —  Fx  sin  a  -f-  Fy  cos  a 

=  0,  (13) 

and  the  normal  component  Fn  by 
Fn  =  Fx  cos  a  +Fy  sin  a 

=  T^r2 —       — :— i ^— '  (14) 


Now  the  charge  per  unit  area  is  easily  shown  to  be 


P  = 


(15) 


4  Trr2  Vl  -  /32  sin2  0  Vcos20  +  (1  -  £2)2  sin20 
Therefore,  as  the  electromagnetic  force  just  inside  the  electron's 

BT=i(F,4-0) 

J   e2  (1-/32)2 


surface  is  zero, 


32 


(16) 


RADIATION  85 

As  this  stress  has  the  nature  of  a  hydrostatic  tension  inde- 
pendent of  the  velocity  of  the  electron,  Poincare  has  been  led 
to  suggest  that  the  electron  may  be  held  together  by  an  equal 
and  opposite  hydrostatic  pressure  of  an  extra-electrodynamical 
character. 

As  the  electron's  velocity  increases  from  0  to  v,  its  volume 
decreases  from  4  3 

f  TTtf 


to  |-7ra3Vl-/32. 

Therefore  the  work  done  against  the  stress  of  (16)  is 


Adding  this  to  (10),  (12)  is  obtained. 

36.  Diffraction  of  X  Rays.  Equation  (3)  of  section  33  shows 
that  in  general  an  electron  will  radiate  energy  whenever  it  is 
accelerated.  However  irregular  its  motion  may  be,  the  radiation 
emitted  may  be  analyzed  by  Fourier's  method  into  a  series 
of  superimposed  simple  harmonic  waves.  Waves  of  a  length 
from  4000  A  to  8000  A  constitute  light  of  the  visible  spectrum, 
whereas  waves  of  a  length  of  the  order  of  1  A  are  called  X  rays. 
These  rays  have  great  penetrating  power,  and  all  attempts  to 
diffract  them  were  unsuccessful  until  Laue  suggested  in  1913 
that  the  distances  between  adjacent  atoms  in  crystals  were  of 
such  a  magnitude  as  to  make  these  substances  suitable  natural 
gratings  for  the  diffraction  of  X  rays.  The  following  theory  is 
presented  very  nearly  in  the  form  given  originally  by  Laue. 

Let  ar  a2,  a3  be  vectors  having  the  lengths  and  directions 
of  the  edges  of  an  elementary  parallelepiped  of  the  crystal. 
Then  if  #,  ?/,  z  are  the  coordinates  of  an  atom  relative  to  an 
origin  0  at  the  center  of  the  crystal, 


y  =  ma,y  +  na^  +  pa^,  (18) 


where  w,  n,  p  are  positive  or  negative  integers.    Let  r  be  the 
distance  of  this  atom  from  the  observer  at  P,  and  R  the  distance 


86        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

of  the  center  of  the  crystal  0  from  P.  These  two  distances 
will,  in  general,  be  so  nearly  equal  that  they  may  be  considered 
the  same,  except  in  so  far  as  the  phase  of  the  radiation  is 
concerned. 

Let  the  incident  radiation  be  plane,  the  direction  cosines  of 
the  wave  normal  being  denoted  by  «0,  /30,  yQ.  Then  if  the  intra- 
atomic  vibrators  are  all  alike,  the  displacement  of  any  vibrator 
at  a  time  t  will  be  given  by  the  real  part  of 


and  the  field  strength  at  P  due  to  this  displacement  by 


A   i^-(r  + 
—  e 
R 

since  the  electric  intensity  at  a  distance  from  the  atom  great 
compared  with  the  wave  length  varies  directly  as  the  acceler- 
ation of  the  vibrator,  which  is  proportional  to  its  displacement, 
and  inversely  as  the  distance.  The  coefficient  A  is  a  function 
of  the  direction  cosines  #,  /#,  7,  of  the  line  OP,  as  well  as  of 

«o'  £>'  V 

The  total  electric  intensity  at  P,  then,  is  equal  to 


«» 

,'  A" 


'  R 


since  r  =  R  —  (xa 

is  a  sufficiently  close  approximation  for  the  exponent. 

Put        F=          aa-  a 


RADIATION  87 

Then,  by  virtue  of  (18), 


^    imF^inO^lpH  (20) 

J       V       T 

Hence  the  intensity  of  the  diffracted  radiation  at  P  is  maxi- 
mum for  those  directions  for  which 


have  their  greatest  absolute  values. 

Suppose  the  illuminated  portion  of  the  crystal  to  be  bounded 
by  planes  parallel  to  the  sides  of  the  elementary  parallelepiped. 
Then  m  varies  from  —  M  to  +  M,n  from  —  N  to  +  TV,  and  p  from 
—  P  to  -f  P,  where  M,  N,  P  are  positive  integers.  Hence 


1-cosMF}. 

The  absolute  value  of  this  expression  is  obviously  a  maximum 
when  F  =  ±2q1ir, 

where  ql  is  a  positive  integer. 

Therefore  the  conditions  for  maximum  intensity  are 
alxa  +  al90  +  algy  =  alxaQ  +  aly/3Q  +  a1370  ±  qfr 
«ax«  +  «-2^  +  «227  =  «»  A  +  a**&*  +  a2,70  ±  qfr       (21) 

«8x«  4-  ^y/3  +  «3*7  =  «3x«0  +  fl8^0  +  ^,70  ±  ^3X' 

where  ^2  and  ^3  are  also  positive  integers. 

The  left-hand  member  of  each  of  these  equations  is  propor- 
tional to  the  cosine  of  the  angle  between  one  of  the  sides  of 
the  elementary  parallelepiped  and  the  diffracted  ray,  while  the 
right-hand  side  is  equal  to  an  integral  number  of  wave  lengths 
plus  a  quantity  proportional  to  the  cosine  of  the  angle  between 
the  same  side  of  this  parallelepiped  and  the  incident  ray.  So 
for  each  value  of  q^  qz,  and  qs  these  equations  define  three  cones 
in  a,  /3,  and  7.  The  loci  of  maximum  intensity  will  be  the 
lines  of  intersection  of  these  cones. 


88        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Consider  the  simple  case  where  a±,  aa,  and  a3  are  mutually 
perpendicular  and  equal.    Choose  axes  parallel  to  the  sides  of 
the  elementary  parallelepiped,  and  consider  an  incident  wave 
advancing  along  the  X  axis.   Then  the  con- 
ditions contained  in  (21)  reduce  to 
aa  =  a  ±  ^X, 
a&  =  ±  Q9\,  (22) 

=  ±     X  ^° 

If  a  screen  is  placed  at  right  angles  to  the 
X  axis  on  the  side  of  the  crystal  opposite 
to  the  source,  the  intensity  of  the  trans- 
mitted radiation  will  be  greatest  along  the 
traces  on  the  screen  of  the  cones  defined 
by  (22).  The  trace  on  the  screen  of  the 
cone  defined  by  the  first  of  these  equations 
is  a  circle,  while  those  defined  by  the  second  and  third  are 
hyperbolae.  The  greatest  intensity  of  all  will  be  produced  at 
those  points  on  the  screen  where  two,  or  better  three,  of  these 
plane  curves  intersect. 

Consider  radiation  incident  on  the  crystal  under  consideration 
at  a  small  glancing  angle  0  (Fig.  14).  If  the  wave  normal  is 
in  the  XY  plane, 


FIG.  14 


and  aa  =  a  sin  6  ±  g^X, 

a/S  =  acosO  ±  g-2X, 
«7  =  ±  &*•• 

Consider  the  reflected  radiation.    For  this 

a  =  —  sin  0, 
ft  =  cos  9, 
7  =  0, 
and  2  a  sin  6  =  ±  g^X, 


(23) 


for  reinforcement. 


CHAPTER  VII 

ELECTROMAGNETIC  FIELDS  IN  MATERIAL  MEDIA 

37.  Fundamental  equations.  Consider  a  medium  composed  of 
positive  and  negative  electrons.  Experimental  observation  of 
the  field  in  such  a  medium  is  limited,  by  the  coarseness  of  avail- 
able instruments,  to  the  investigation  of  the  average  values  of 
the  electric  and  magnetic  intensities.  These  average  values  are 
denned  in  the  following  way.  Divide  the  medium  up  into  fixed 
elements  of  volume  T  so  large  that  each  contains  very  many 
electrons,  but  yet  so  small  that  no  measuring  instrument  which 
may  be  used  to  investigate  the  field  can  detect  a  variation  of 
electrical  properties  from  one  point  to  another  in  any  one  of 
these  elements.  Then  the  average  value  at  any  point  of  a 
scalar  <f>  which  depends  upon  the  state  of  the  electrons  in  the 
medium  is  defined  by  _ 


where   the  integral  is  taken  through  the  volume   element  T. 
Similarly,  the  average  value  of  a  vector  V  is  defined  by 


In  the  absence  of  an  impressed  field,  each  volume  element  of 
the  medium  will  be  supposed  to  contain  equal  numbers  of  posi- 
tive and  negative  electrons  moving  about  in  a  fortuitous  manner. 
Consequently  the  average  charge  and  average  current  will 
vanish.  In  the  presence  of  a  field,  however,  the  electric  inten- 
sity may  cause  electrons  of  opposite  sign  to  be  displaced  in 
opposite  directions,  with  the  result  that  the  average  density  of 
charge  may  no  longer  be  everywhere  zero.  In  the  same  way, 
the  magnetic  intensity  may  orient  intra-atomic  rings  of  electrons 
in  such  a  way  as  to  produce  an  average  current  different  from 

89 


90        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

zero.    Analytical  difficulties  involved   in  discontinuities  in  the 
medium  may  be  avoided  by  imagining  every  charged  surface  to  be 
replaced  by  a  region  of  finite  but  very  small  thickness,  in  which 
p  varies  rapidly  but  continuously  from  one  value  to  another. 
As  the  elements  of  volume  r  are  fixed, 


dx      rjr  dx 


-s* 


etc.,  and  similarly  for  the  vector  function  V. 

To  find  the  average  value  of  /o,  consider  a  volume  element  r 
of  dimensions  A#,  Ay,  A«.  In  the  time  dt  the  charge  entering  T 
through  those  sides  of  this  volume  element  which  are  perpen- 
dicular to  the  X  axis  is 

~pvxt±ykzdt  —  |  ~pvx  +  —  (fvx)  Az  J- 

o 

=  -  g^G™ 
Therefore,  takmg  the  six  sides  of  r  into  account, 


%' 

Put  ( 


=  Cpvdt. 


Then  p  =  -  V-Q,  (1) 

and  v  =    - 


Hence,  for  a  material  medium,  the  equations  of  the  electro- 
magnetic field  given  in  section  26  take  the  following  form: 

V-(I  +  Q)=0,  (3)  V-H  =  0,  (5) 

VxE  =  --g,     (4)  VXH  =  -(E  +  Q).  (6) 


MATERIAL  MEDIA  91 

The  method  used  in  deriving  these  equations  has  not  been 
such  as  to  limit  their  applicability  to  media  which  are  homo- 
geneous or  isotropic.  They  apply  to  all  media  made  up  of 
electrons  whatever  their  nature.  Moreover,  as  two  media  in  con- 
tact may  be  considered  equivalent  to  a  single  non-homogeneous 
medium  containing  a  thin  transition  layer  in  which  the  prop- 
erties of  the  medium  change  rapidly  but  continuously,  these 
equations  may  be  applied  in  the  region  of  contact. 

In  order  to  determine  the  relation  between  Q  on  the  one 
hand  and  E  and  H  on  the  other,  it  is  necessary  to  consider  the 
motion  of  individual  electrons  in  the  medium.  These  electrons 
consist  of  two  classes  :  (a)  the  free  electrons,  which  move  among 
the  atoms,  and  (b)  the  bound  electrons,  whose  displacements  are 
limited  by  the  boundaries  of  the  atoms  to  which  they  belong. 

As  an  electron  is  of  very  small  dimensions  compared  with  an 
atom,  the  number  of  times  a  free  electron  collides  with  another 
electron  is  negligible  compared  with  the  number  of  times  it  col- 
lides with  an  atom.  Hence  if  an  electron  strikes  against  an  atom 
v  times  a  second,  its  average  drift  velocity  is  given  by 


where  f  is  the  average  acceleration  produced  by  the  intensities 
E!  and  Hx  due  to  all  charges  other  than  that  on  the  electron 
under  consideration.  Now 

mi  = 


approximately,  and  therefore  if  v  is  small  compared  to  c  the 
current  due  to  the  free  electrons  is  given  by 


where  N  is  the  number  of  free  electrons  per  unit  volume. 

Describe  a  sphere  of  volume  1/N  around  the  electron  under 
consideration.  Then  if  E2  is  the  average  electric  intensity  within 
this  sphere  due  to  this  electron, 

£,=!-!,. 


92        AN  INTRODUCTION  TO  ELECTRODYNAMICS 

In   general   E2  is   small   compared   to   the    other   quantities 
involved  in  this  equation.    Hence  (7)  may  be  written 


(8) 

where  C  is  the  conductivity. 

A  bound  electron  is  supposed  to  be  held  in  the  atom  to 
which  it  belongs  by  a  force  of  restitution  proportional  to  its 
displacement  R  from  the  center  of  the  atom.  In  addition,  it 
is  supposed  to  be  subject  to  a  frictional  resistance  proportional 
to  its  velocity.  Hence,  if  E±  is  the  electric  intensity  and  Ht 
the  magnetic  intensity  due  to  all  causes  external  .to  the  atom 
under  consideration,  the  equation  of  motion  of  a  bound  electron 
inside  this  atom  is 


where  those  terms  in  the  dynamical  reaction  other  than  the  one 
due  to  the  mass  have  been  omitted  as  negligible. 

Put  21  =  -, 


m 


If*  =  ^ , 
Ko  —  — 

m 


Then  the  equation  of  motion  becomes 

R  +  2 IR  +  &02R  =  -  (E±  +  -  R  X  HA.  (9) 

m  I         c  ) 

The  displacement  R  satisfying  this  equation  consists  of  the 
solution  R0  of  the  complementary  equation 

R0+  2ZR0+  &02R0=  —  R0  x  H,  (10) 

//to 

added  to  a  particular  solution  R^  of  the  original  equation. 

Obviously  R^  represents  the  part  of  the  displacement  pro- 
duced by  the  electric  field  E1?  while  R0  is  the  part  due  to  the 


MATERIAL  MEDIA  93 

natural  motions  of  the  electrons  inside  the  atom  as  modified  by 
the  presence  of  the  magnetic  field  Hr  Since  Rp  is  in  general 
negligible  compared  to  c,  the  particular  solution  may  be  obtained 
to  a  sufficient  degree  of  accuracy  from  the  equation 


(11) 


Now  if  there  are  N  atoms  per  unit  volume,  and  each  of  these 
contains  n  electrons,  the  current  produced  by  the  bound  electrons 

is  given  by  -^—          -^~ 

pv  =  NneRp  +  NneRQ. 

While  eRQ  may  vary  greatly  from  one  electron  to  the  next, 
it  is  evident  that  eRp  will  have  more  or  less  the  direction  of 
the  average  electric  intensity.  Consequently  the  contribution  to 


due  to  the  entrance  of  new  electrons  into  the  volume  element  r 
through  which  the  average  is  being  taken  will  be  negligible  com- 
pared to  the  part  of  this  quantity  dependent  upon  the  electrons 
already  inside  this  region.  Therefore 


and  pv  =  NneRp+NneRQ.  (12) 

If  the  electrons  inside  the  atom  are  subject  to  different  re- 
storing and  resisting  forces,  this  equation  must  be  replaced  by 


pv  =  Nn+Nn  •        (13) 

i  t 

where  nt.  denotes   the  number  of   electrons   of   type   i   inside 
each  atom. 

The  electric  polarization,  or  electric  moment  per  unit  volume,  is 

defined  as  „          -^=- 

P==NneRp, 

or,  if  each  atom  contains  more  than  one  type  of  electron, 


94        AN  INTRODUCTION  TO  ELECTRODYNAMICS 
Hence  (13)  may  be  written 

pv=P+N'£nieR9.  (14) 

• 

At  first  sight  it  would  seem  as  though  eRQ  must  always  vanish. 
In  a  magnetic  substance,  however,  this  is  not  necessarily  the 
case,  as  will  now  be  shown. 

Suppose  that  for  certain  types  of  electrons  the  damping 
coefficient  I  in  equation  (10)  is  zero,  and  that  the  motion  is 
constrained  to  a  plane.  Then  motion  in  a  circle  with  con- 
stant frequency  is  a  solution  of  this  equation.  A  substance 
each  of  whose  atoms  contains  one  or  more  rings  of  electrons 
revolving  about  their  respective  axes  is  said  to  be  magnetic. 
The  electrons  in  each  ring  will  be  assumed  to  be  evenly  spaced, 
and  to  have  a  constant  angular  velocity  Q  about  the  axis 
of  the  ring  so  long  as  the  external  magnetic  field  remains 
unchanged. 

In  the  presence  of  a  magnetic  field  each  electron  will  be 
subject  to  a  force  which  will  tend,  in  general,  to  change  the 
radius  RQ  of  the  ring  and  the  frequency  of  revolution.  More- 
over, the  ring  as  a  whole  will  be  acted  on  by  a  torque  which 
will  tend  to  orient  it.  The  force  on  an  electron  resulting  from 
the  magnetic  intensity  Hx  due  to  all  causes  external  to  the 
atom  in  which  it  lies  is  given  by 

K  =  -(vxH1) 

0 

=  -(flxR0)xH1, 

c 

as  R0  is  measured  from  the  center  of  the  ring.    The  torque  about 

this  point  is 

G=]TR0xK 


for  the  entire  ring.    The  form  of  this  expression  shows  that  G 
is  perpendicular  to  ft.    Denote  by  <f>  the  angle  which  R0  makes 


MATERIAL  MEDIA 


95 


with  the  plane  of  ii  and  H±.  Then  if  6  is  the  angle  between  fl 
and  H1?  the  component  G^  of  the  torque  parallel  to  the  plane 
of  these  two  vectors  is 

6rM  =  -  ^R££IH  sin#  sin  <£  cos  <f> 
c  ** 

=  0, 
and  the  component  GL  at  right  angles  to  this  plane  is 


where  n  is  the  number  of  electrons  in  the  ring.    The  quantity 


is  called  the  magnetic  moment  of  the  ring.    Therefore  the  torque 
is  given  by  G  =  MxH1.  (15) 

Denote  by  dN  the  number  of  atoms  per  unit  volume  in  which 
the  axes  of  the  rings  of  electrons  have  a  given  direction.  Then 
the  intensity  of  magnetization,  or  magnetic 
moment  per  unit  volume,  is  denned  as 


•A' 


or,  if  each  atom  contains  more  than  one      O      P    Q  x%x 

ring,  I  =  2VVM..  *%XV> 

*f     l'  FIG.  15 

Consider  a  ring  of  electrons  whose  center  is  located  on  the 
JFaxis  (Fig.  15)  at  a  distance  p  to  the  right  of  the  origin.  Denote 
by  6  the  angle  which  M  makes  with  the  Xaxis,  and  by  (f>  the 
angle  which  the  MX  plane  makes  with  the  XY  plane.  It  is 
desired  to  find  the  value  of  ^  • 


96        AN  INTRODUCTION  TO  ELECTKODYNAMICS 

due  to  electrons  in  this  ring  which  lie  to  the  right  of  the  YZ 
coordinate  plane.    Evidently  this  sum  will  differ  from  zero  only 
if  the  ring  is  cut  by  this  plane  ;  that  is,  if 
-  RQ  sin0  <  p  <  RQ  sin0. 

Moreover,  it  is  obvious  that  this  sum  will  have  no  component 
in  the  X  direction.  If  i/r  is  the  angle  between  R0  and  the  inter- 
section QP  of  the  plane  of  the  ring  with  the  MX  plane, 


=  sn 


integrated  from 

to  cos 

mi        r  /NT^   T\  \       nefl  V R*  sin2$ — p'2   .     . 

Therefore       (  Z_,eRn  )„  = — *-  sin  6. 

TT  sin  6 

Summing  up  over  a  prism  of  unit  cross  section  extending  from 

p  =  —  RQ  sin  0 

to  p  =  fi0  sin  6, 

the  y  component  of  the  current  to  the  right  of  the  YZ  plane  due  to 
all  rings  which  are  cut  by  a  unit  area  of  this  plane  is  found  to  be 


=  c  I  Msi 

-.fit 


Msin  9  sin 


Similarly, 

and,  in  general,  eo=  cl  x  n' 

where  n  is  a  unit  vector  normal  to  the  surface  on  the  positive 

side  of  which  the  current  is  to  be  computed. 


MATERIAL  MEDIA  97 

Consider  a  volume  element  r  of  dimensions  A#,  A^,  Az.  The 
current  contained  in  it  on  account  of  those  rings  of  electrons 
which  are  cut  by  the  sides  perpendicular  to  the  X  axis  is 

cl  x 


Therefore,  taking  the  six  sides  of  r  into  account, 

JVw«R0=cVxI,  (17) 

or,  if  each  atom  contains  more  than  one  ring  of  electrons, 

2V2Jwt.gR0=cVxL  (18) 

i 

Hence  the  entire  current  due  to  bound  electrons  is  given  by 
pv  =  P  +  cVxl.  (19) 

Consider  a  medium  in  which  there  may  exist  currents  due 
both  to  conduction  by  free  electrons  and  to  the  displacement 
of  bound  electrons.  Then  it  follows  from  (8)  and  (19)  that 

Q  =  CE  +  P  +  cV  x  I,  (20) 

and  equation  (6)  becomes 

V  x  (H  -  I)  =  i(I  +  P  +  CE).  (21) 

In  accord  with  the  usual  practice  put 


and  B  =  H. 

The  vector  D  is  called  the  electric  displacement,  and  B  the 
magnetic  induction.  The  average  magnetic  intensity  due  to  all 
causes  other  than  the  intensity  of  magnetization  I  of  the  medium 
is  usually  denoted  by  H.  As  this  letter  has  been  used  with 
another  meaning  in  the  preceding  pages,  this  quantity  will  be 
designated  by  L.  Then,  omitting  strokes,  equation  (6)  gives 


and  comparison  with  (21)  shows  that 

L=B-I, 


98    AN  INTRODUCTION  TO  ELECTRODYNAMICS 

since  L  vanishes  with  B  and  I.  As  B  is  the  total  average  mag- 
netic intensity,  I  represents  the  average  magnetic  intensity 
produced  by  the  magnetization  of  the  medium.  Equation  (21) 
may  now  be  written 

VxL  =  -(D  +  CE).  (22) 

Returning  to  (20),  it  follows  that 


and  V-Q  =  C  Cv-Edt  +V-P,  (23) 

as  Q  and  P  vanish  everywhere  when  no  external  field  is  present. 
Therefore,  omitting  strokes,  equation  (3)  becomes 


V«D  =  -  C  I  V-Ecft.  (24) 


Hence  the  equations  of  the  electromagnetic  field  in  a  medium 
containing  both  free  and  bound  electrons  take  the  form 

V-V  =  -C  Cv-Edt,     (25)  V-B  =  0,  (27) 

VxE  =  --B,  (26)  VxL  =  -(D  +  <7E),     (28) 

c  c 

when  the  field  is  investigated  from  the  macroscopic  point  of  view. 
As  already  noted,  these  equations  apply  whether  or  not  the 
medium  is  homogeneous  or  isotropic,  and  they  are  valid  in  the 
region  of  contact  of  two  media. 

If,  in  addition  to  the  processes  of  conduction  and  displace- 
ment, charges  p  and  currents  pv  are  produced  by  convection 
through  the  medium,  the  electromagnetic  equations  for  this  most 
general  case  assume  the  form 

V-D  =  -(7  Cv-Edt+p,    (29)       V-B  =  0,  (31) 

VxE=-ifi,  (30)     VxL=^(I)  +  CfE+/0v).     (32) 


MATERIAL  MEDIA 


99 


38.  Specific  inductive  capacity.  In  order  to  obtain  the  relation 
between  P  and  E,  and  hence  between  D  and  E,  it  is  necessary  to 
solve  equation  (11)  of  the  preceding  section.  The  following  dis- 
cussion will  be  confined  to  the  case  of  simple  harmonic  fields, 
a  steady  field  being  considered  as  a  simple  harmonic  field  of 
zero  frequency.  The  electric  and  magnetic  intensities  are  given 
by  the  real  parts  of  the  following  complex  quantities, 


and 


d 


Hence  the  solution  of  (11)  is 


e 
m 


and  P  is  given  by      P  = 

g  J 


Nn- 
m 


(33) 


To  find  the  relation  between  Et  and  E,  describe  about  the 
center  0  (Fig.  16)  of  each  atom  a  sphere  of  volume  1/N.  Then 
if  E2  is  the  average  electric  intensity 
within  one  of  these  spheres  due  to  the 
electrons  which  it  contains, 


Evidently  E2  has  the  direction  of 
the  displacement  R^  caused  by  the  ex- 
ternal field  Et,  and  its  magnitude  is 
given  approximately  by 


FIG.  16 


=  Nn       -r—2  cos  6  (2  Trr2  sin  0d0dr), 


100      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

where  6  is  the  angle  which  the  radius  vector  makes  with 
As  r  goes  from  zero  to 


RA- 


it  follows  that  E2  =  -  1  NneRp.  (34) 

Therefore  E±  =  E  +  J  P, 

Nn- 

/yy) 

and  (33)  becomes  P=  -  -  -  —  —  -  ;I,  (35) 

K  —  CO2—  2lO)l  , 

1         e2 

where  &2  =  k2  —  -  Nn  —  • 

3        m 

Nn  — 

m 
Put  e 


k2  -co2—  2  icol 
Then,  omitting  strokes, 

P  =  eE,  (36) 

and  D  =  (l+e)E.  (37) 

The  specific  inductive  capacity  K  is  defined  as  the  ratio  of  the 
displacement  D  to  the  electric  intensity  E.    Hence 

K  =  1-1-  e 


Nn- 


k2-  co2-  2  icol 
or,  if  each  atom  contains  more  than  one  type  of  electron, 


(38) 


-          (39> 

In  the  case  of  a  steady  field, 


if  there  is  only  one  kind  of  electron  in  each  atom,  and 

Nne2  _  K  —  1 
3  kQ2m  ~  K  -f  2 ' 


(41) 


MATERIAL  MEDIA    , 

The  number  of  atoms  per  unit  volume  varies  directly  with 
the  density  d.    Therefore  n 


should  be  a  constant.    This  relation  has  been  verified  for  many 
gases. 

The  force  of  restitution  in  a  polarized  atom  may  be  easily 
evaluated  if  n  positive  and  n  negative  electrons  are  assumed 
to  be  distributed  uniformly  over  equal  and  coincident  spheres. 
If  the  center  of  one  of  these  spheres  remains  stationary,  while 
that  of  the  other  is  displaced  a  distance  R^,  the  force  drawing 
them  together  is  approximately 


ne 


=  1—7? 

~3    v       p' 

where  v  is  the  volume  of  a  sphere.    Hence  the  force  on  each 

electron  is  .,      2 

1  ne 

and  k%  =  -  — . 

o  vm 

Substituting  in  (41),  it  is  found  that 

(43) 


is  the  portion  of  the  volume  under  consideration  which  is 
actually  occupied  by  the  atoms.  If  an  estimate  of  N  has  been 
obtained  from  other  sources,  the  atomic  volume  v  may  be  com- 
puted. In  this  way  the  radius  of  an  atom  is  found  to  be  of 

the  order  of  ,^AN 

(10)~8cm. 

39 .  Magnetic  permeability.  It  has  been  pointed  out  in  section  3  7 
that  a  magnetic  field  tends  both  to  change  the  radius  and  the 
frequency  of  revolution  of  the  ring  of  electrons  inside  a  magnetic 


TO  ELECTRODYNAMICS 

atom,  and  to  orient  this  ring.    To  determine  the  magnitude  of 
the  first  effect,  integrate  both  sides  of  the  equation 


over  the  surface  bounded  by  a  ring  of  electrons.    Then 
fvxE^o^-  !-  fll^o-. 

But  by  Stokes'  theorem 

CvxE^d(T=  /X-dX, 

where  the  right-hand  member  is  integrated  around  the  ring. 
Hence  the  work  done  on  the  ring  when  the  magnetic  field  is 
increased  by  c?Ht  is 


=  -M-dRi.  (44) 

Now  the  increase  in  the  kinetic  and  potential  energies  of  the 
electrons  in  the  ring  is 

dU  =  nm  (ydv  +  k^R0dRQ) 
£l  +  2 


as  k0=fl. 


But  dM  =          (^^n  4- 

Therefore  dU  =  ^^  dM.  (45) 

£ 

Equating  this  to  the  expression  for  the  rate  at  which  work 
is  done  on  the  ring, 

-dH,.  (46) 

x  v     } 


MATERIAL  MEDIA  103 

Let  a  be  the  angle  which  M  makes  with  c?H1.  If  initially  the 
axes  of  as  many  atomic  rings  pointed  in  one  direction  as  in  any 
other,  dl  has  the  direction  of  dlL^  and  is  given  in  magnitude  by 

dl=  I  dMcosadN 

_e_MdH1 
~  2  mile 

NeM  ^  . 

= d-H.  .  v**y 

To  find  the  relation  between  6?Ht  and  c?H,  describe  about  the 
center  of  each  atom  a  sphere  of  volume  1/jV". .  Then  if  Ha  is  the 
average  magnetic  intensity  within  one  of  these  spheres  due  to 
the  electrons  which  it  contains, 


But  it  has  been  shown  that  the  average  magnetic  intensity  H2 
due  to  rings  of  electrons  is  equal  to  the  intensity  of  magneti- 
zation I.  Therefore  as  _  = 

-D  —  Jri , 

it  follows  that  Hx  =  B  —  I 

=  L, 

and  dl  =  — dL.  (48^) 

6m£lc 

The  permeability  /JL  of  the  medium  is  defined  as  the  ratio  of 
the  magnetic  induction  B  to  the  external  field  strength  L. 
Therefore  in  the  case  under  consideration 

NeM 


The  effect  under  discussion  is  known  as  diamagnetism,  and  is 
characterized  by  a  value  of  the  permeability  less  than  unity. 
It  is  shown  in  the  greatest  degree  by  bismuth. 

The  tendency  of  a  magnetic  field  to  orient  a  ring  of  electrons 
will  be  considered  next.  The  torque  on  such  a  ring  has  already 
been  shown  to  be  ~  .*,  _ 

w  =  JVL  X  H. 


104      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Let  H  be  the  average  total  magnetic  intensity  inside  a  sphere 
of  volume  \/N  described  about  the  center  of  the  atom  under 
consideration,  and  let  H2  be  the  average  magnetic  intensity 
within  this  sphere  due  to  the  electrons  which  it  contains.  Then 


where  H2  evidently  has  the  direction  of  M.    Therefore 

G  =  M  x  H.  (50) 

Consider  a  ring  of  electrons  which  is  rotated  from  a  position 
where  M  is  perpendicular  to  H  to  one  where  M  makes  an  angle  a 
with  H.  The  potential  energy  acquired  is 

sinada 


=  MH  f 


=  —  MH  cos  a 

=  -M-H.  (51) 

Therefore  if  the  tendency  of  the  magnetic  field  to  bring  all 
the  atomic  rings  into  line  is  opposed  only  by  the  disorganizing 
effect  of  thermal  agitation,  the  number  of  rings  per  unit  volume 
whose  axes  make  angles  with  H  between  a  and  a  -+-  da  is  given  by 


dN=\Ae 

MHCOB* 

=  ^Ae  *kT    sin  a  da, 

where   T  is  the   absolute   temperature   and  1  JcT  the   average 
kinetic  energy  associated  with  each  degree  of  freedom. 
Integrating,  the  constant  A  is  found  to  be  given  by 

NX 

•A.  —  —   -      » 
smn  x 

MH 
*" 


MB 


MATEKIAL  MEDIA  105 

The  magnetic  moment  per  unit  volume  has  the  direction  of  B 
and  is  given  in  magnitude  by 


=  I 


McosadN 

(52) 


x      x 


As  B  increases  I  does  also,  approaching  the  saturation  value 
NM  for  large  values  of  the  field.    For  small  fields 

I=-NMx 
o 


showing  that  the  intensity  of  magnetization  is  proportional  to 
the  strength  of  the  field.    The  permeability  is  given  by 


The  effect  under  discussion  is  known  as  paramagnetism,  and 
is  characterized  by  a  value  of  the  permeability  greater  than 
unity.  It  is  shown  to  an  exceptional  degree  by  iron,  in  which 
the  effect  is  given  the  special  name  of  ferromagnetism.  The  theory 
given  above  cannot  be  considered  as  more  than  very  roughly 
approximate  to  the  facts,  especially  as  it  gives  no  explana- 
tion of  hysteresis,  or  the  lagging  of  the  magnetization  behind 
the  field. 

40.  Energy  relations.    From  (30)  and  (32)  it  follows  that 

L-V  x  E  -  E-V  x  L  =  -  i  (E-D  +  L-B)  -  -  (CE2  +  pE-v). 

But  L-V  x  E  -  E-V  x  L  =  V-  (E  x  L). 

Hence 

=  0. 


106      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Integrating  over  any  arbitrarily  chosen  portion  of  space,  and 
applying  Gauss'  theorem  to  the  second  term, 

j  C\(icE2+nL^dT+c  C(ExL)-d<y+(C£2+PE.v)dT=0.     (55) 

The  third  term  of  this  expression  measures  the  rate  at  which 
work  is  done  by  the  electromagnetic  field  on  the  conduction 
and  convection  currents  in  the  medium.  Following  the  line  of 
reasoning  pursued  in  section  28,  the  conclusion  is  reached  that 
the  first  term  represents  the  rate  of  increase  of  energy  of  the 
field,  and  the  second  the  flux  of  energy  through  the  surface 
enveloping  the  field.  The  forms  of  these  expressions  suggest  that 


is   to   be   considered   as   the   electromagnetic  energy  per  unit 

volume,  and  ,_      _, 

s  =  c  (E  x  L) 

as  the  flow  of  energy  per  unit  cross  section  per  unit  time. 

41.  Metallic  conductivity.  In  developing  the  electron  theory 
of  metallic  conduction,  the  atoms  in  a  metal  may  be  treated  as 
immobile  compared  with  the  electrons.  Conduction  currents  of 
electricity  in  a  metal  will  be  supposed  to  be  due  entirely  to  the 
drift  velocity  of  the  free  electrons  in  the  direction  of  an  im- 
pressed electric  field,  and  heat  conduction  will  be  attributed  to 
the  transport  of  energy  by  these  electrons  from  one  atom  to 
another  in  the  direction  of  the  temperature  gradient. 

If  u  is  the  average  velocity  of  an  electron  due  to  thermal 
agitation,  and  I  the  average  path  described  between  collisions 
with  atoms,  the  number  of  times  an  electron  strikes  an  atom 
per  second  is  given  by 


and  equation  (8)  for  the  conduction  current  becomes 


MATERIAL  MEDIA  107 


As  i  mu2  =  I  kTj 

the  electrical  conductivity  is  given  by 

C=—. 
QkT 


(56) 


In  determining  the  heat  conductivity,  take  the  X  axis  in  the 
direction  of  the  temperature  gradient.  Consider  an  electron 
which  is  just  about  to  collide  with  an  atom  at  a  distance  x  from 
the  origin.  This  atom  will  have,  on  the  average,  a  kinetic  energy 

|^ 
but  the  electron  will  have  the  energy 


of  the  last  atom  with  which  it  collided,  where  6  is  the  angle 
which  the  electron's  path  makes  with  the  X  axis.  During  the 
collision  the  electron  will  come  into  thermal  equilibrium  with 
the  atom,  giving  to  the  latter  an  amount  of  energy  equal  to 

37dT7 

--&  —  -Zcosfl. 
2     dx 

Now  the  number  of  electrons  per  unit  volume  whose  paths 
make  angles  between  6  and  6  +  dO  with  the  X  axis  is 


and  the  number  of  these  which  pass  through  unit  area  at  right 
angles  to  the  X  axis  in  unit  time  is 

Nu  cos  e  sin  6d6. 


Hence  the  -flux  of  energy  is 


—  f  " 
dxJ0 


-  K—  =  --  Nluk  —       cos'tf  sin  OdO 

dx 


2  dx 


giving  for  the  thermal  conductivity 

'   K=  \Nluk.  (57) 


108      AN  INTRODUCTION  TO  ELECTRODYNAMICS 
The  ratio  of  thermal  to  electrical  conductivity, 

(58) 


varies  directly  with  the  absolute  temperature  and  is  the  same 
for  all  metals  at  a  given  temperature.  Its  value  for  any  tem- 
perature depends  only  upon  the  universal  constants  k  and  e, 
whose  values  may  be  determined  from  experiments  having  no 
connection  with  the  metallic  conductor  in  question.  The  ratio 
of  the  conductivities  as  thus  computed  is  in  fair  agreement 
with  the  ratio  as  determined  directly  by  experiment. 

In  obtaining  the  expressions  for  the  conductivities  given 
above,  use  has  been  made  of  the  average  velocity  of  thermal 
agitation  and  the  average  length  of  path  between  successive 
collisions.  A  more  exact  calculation  gives  a  slightly  different 
numerical  coefficient  for  the  ratio,  but  one  which  shows  rather 
worse  agreement  with  the  experimental  value  of  this  quantity. 

42.  Reduction  of  the  equations  to  engineering  form.  If  free 
charges  and  currents  —  either  conduction  or  convection  —  are 
present  in  a  material  medium,  the  equations  (29)  to  (32), 
section  37,  of  the  electromagnetic  field  may  be  written  in 
the  form 


(59)  V-B  =  0,  (61) 


0 


(60)  VxL  =       D  +  j,     (62) 


where  J  is  the  current  density  ;  that  is,  the  current  per  unit  cross 
section.    Moreover,  j=<7E+/>v,  (63) 


=  *E,  (64) 


(65) 

The  quantities  involved  in  these  equations  are  measured  in 
Heaviside-Lorentz  units.    If  the   same  quantities  as  measured 


MATERIAL  MEDIA  109 

in  electromagnetic  units  are  designated  by  letters  with  the  sub- 
script m,  and  as  measured  in  electrostatic  units  by  letters  with 
the  subscript  *,  then 

p  =  <?  V4  7rpm  =  V4  7T/os, 


P  =  c  V4  7rPm  =  V4  7rPs, 


^-B. 


E  =  — -=  Em  =     , Es, 


4-7T 

1  _E  -    — 

~    "l~V4"7r 

L  =  -^=Lm=— 7=L, 


Therefore,  in  electromagnetic  units  the  field  equations  take 
the  form 


.Jmdt,     (66)  V.Bm=0,  (68) 

VxEm=-Bm,  (67)         VxLm=47r(Dm+JJ,  (69) 

where  T   =  C  E   +  p  v,  (TO') 

•'/n  ni/Ti'iiu'  \         s 


^^  (71) 


=  ^Lm-  (72) 
In  electrostatic  units  these  equations  are 

V-D,=  ri,'  (73)  V-BS=0,  (75) 

VxEs=-Bs,       (74)  VxLs=  4^(6,+!,),  (76) 


110      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

where  the  charge  accumulated  at  any  point  has  been  denoted  by 
p(  instead  of  r 

-jv-JA 


and 


<78> 


2 


In  practical  applications  of  electrodynamics  these  equations 
are  generally  made  use  of  in  integral  form.  For  instance,  con- 
sider a  small  charge  e  permanently  at  rest  at  the  center  of  a 
sphere  of  radius  r.  Integrating  (73)  throughout  the  region 
enclosed  by  the  spherical  surface, 


=  e. 


But  f  V-  D/?T  =  TDS-  d<r 

=  4  7rr2Ds 
by  Gauss'  theorem.    Hence 


D-     "> 
^'~4,n-2 


and,  by  virtue  of  (78),       ^=^*  (80) 

Again,  integrating  (67)  over  a  surface, 


MATERIAL  MEDIA  111 

But,  by  Stokes'  theorem, 


where  the  line  integral  is  taken   along  the  boundary  of  the 
surface.    Therefore 

(81) 


which  is  the  usual  form  of  Faraday's  law  of  current  induction. 
Similarly,  (69)  leads  to 


j*Lm-d\=  4  ^(D.+jJ-dr,  (82) 

which  is  the  form  in  which  Ampere's  law  is  generally  expressed. 


CHAPTER  VIII 

ELECTROMAGNETIC  WAVES  IN  MATERIAL  MEDIA 

43.  Isotropic  non-conducting  media.  For  wave  lengths  long 
compared  with  atomic  dimensions,  the  electromagnetic  field  is 
specified  by  equations  (25)  to  (28)  of  the  last  chapter.  For 
non-conducting  media  these  take  the  form 

V-D  =  0,  (1)  V-B=0,  (3) 

VxE=-ifi,         (2)  VxL=-D,       (4) 

c  c 

where  D  =  /eE 


Eliminating  B  as  in  section  29,  it  is  found  that 

V-VB-^E  =  O,  (5) 

or,  eliminating  D,          V»VL  -  ^  L  =  0.  (6) 

C 

These  are  equations  of  a  wave  travelling  with  velocity 

(7) 


The  permeability  of  all  substances  is  practically  unity  for 
frequencies  as  great  as  that  of  light.  Hence  for  light  waves  it 
is  permissible  to  write  c 

V/c 

The  index  of  refraction  n  of  a  medium  is  defined  as  the  ratio 
of  the  velocity  of  light  in  vacuo  to  that  in  the  medium. 

Therefore  ri*  =  tc,  (9) 

and,  as  was  shown  in  section  38, 


m  ~*  kf  —  a)  —  2  ^Wi 
112 


ELECTROMAGNETIC  WAVES  113 

In  the  case  of  a  plane  wave  it  is  often  convenient  to  make 
use  of  the  wave  slowness  S  in  place  of  the  wave  velocity  V. 
This  quantity  is  defined  as  a  vector  having  the  direction  of  the 
wave  velocity  but  equal  in  magnitude  to  its  reciprocal.  Hence 
the  index  of  refraction  may  be  defined  as  the  ratio  of  the  wave 
slowness  S  in  the  medium  to  the  wave  slowness  SQ  in  vacuo. 
The  electric  intensity  in  the  case  of  a  simple  harmonic  plane 
wave  advancing  in  a  direction  making  angles  #,  /3,  7  Avith  the 
X,  Y,  Z  axes,  may  be  expressed  in  terms  of  the  wave  slowness 
by  the  real  part  of 


which  is  far  more  compact  than  the  equivalent  expression 

(x  cos  a.  +  y  cos  ft  +  z  COB  -y       \ 

1(1)  I £  I 

involving  the  wave  velocity. 

Equation  (8)  shows  that  the  wave  slowness  is  given  by 

S=S^K,  (11) 

where  K  is  in  general  complex.  To  show  the  significance  of  a 
complex  wave  slowness,  put 

S~S'+ e'S", 
where  Sf  and  S"  are  real.    Then 

E  =  EQe~  wS"*r  &•  cs/'r  ~  °,  (1 2) 

showing  that  the  imaginary  part  of  the  wave  slowness  measures 
the  damping  of  the  wave  as  it  progresses  into  the  medium, 
whereas  the  real  part  determines  the  actual  velocity  of  propa- 
gation. The  same  statement  applies  to  a  complex  index  of 
refraction.  If  _, 

where  v  and  x  are  real,  it  follows  from  (10)  that 

^V        rc,(£.2-o>2)  -q 

Z,7T^ ,2^2    .      A     ,.272*  \^6) 


114      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

For  the  portion  of  the  spectrum  to  which  these  expressions 
are  to  be  applied  it  will  be  assumed  that 


Then 


approximately.    Consider  the  denominator 

(£*_«*)«  +4  flfl* 

of  one  of  the  terms  in  the  right-hand  member  of  this  equation. 
Except  for  the  region  where  co  is  very  nearly  equal  to  kt  the 
second  term  of  this  denominator  is  negligible  compared  to  the  first. 
Within  this  region  the  first  is  negligible  compared  to  the  second. 
So  if  v2  is  plotted  against  o>2  in  a  region  extending  from  o>2  to  o>2, 
such  that  &j  lies  between  o>1  and  o>2,  and  &2,  &3,  etc.  lie  outside 
this  range,  2 


except  in  the  neighborhood  of  &2,  where 


Plotting  each  term  separately,  the  dotted  curves  of  Fig.  17 
are  obtained.  Adding  these  curves,  the  full  line  curve  is  found 
to  give  the  relation  between  z/2  and  a>3.  The  portion  of  the  curve 
from  A  to  B  corresponds  to  regular  dispersion,  the  index  of 
refraction  increasing  as  the  frequency  becomes  greater,  whereas 
the  part  BC  accounts  for  the  anomalous  dispersion  observed  in 
the  neighborhood  of  an  absorption  band.  It  must  be  remem- 
bered that  the  index  of  refraction  refers  always  to  the  phase 
velocity  of  a  train  of  waves.  Hence  the  fact  that  this  index 
becomes  less  than  unity  on  the  short  wave  length  side  of  an 


ELECTROMAGNETIC  WAVES 


115 


absorption  band  may  not  be  adduced  to  show  that  an  electro- 
magnetic signal  can  be  despatched  with  a  velocity  greater  than 
that  of  light  in  vacua.  In  fact  Sommerfeld  has  shown  that  the 


forerunners  of  a  limited  train  of  waves  travel  with  the  same 
velocity  c  in  all  media  and  are  undeviated  when  they  pass  from 
one  medium  into  another. 

44.  Anisotropic  non-conducting  media.  For  wave  lengths  long 
compared  with  atomic  dimensions,  the  first  four  equations  of  the 
electromagnetic  field  are  the  same  as  for  isotropic  media.  The 
relation  between  D  and  E,  however,  is  different,  as  the  atoms  in  a 
body  which  is  not  isotropic  must  be  supposed  to  exert  different 
restoring  forces  in  different  directions.  Hence  equation  (36), 
section  38,  for  the  polarization  must  be  replaced  by  the  more 
general  relation  P^  =  ^  +  ^  +  ^ 

and  similar  expressions  for  Py  and  Pz.  In  the  following  dis- 
cussion damping  will  be  assumed  negligible.  Therefore  the 
coefficients  of  the  components  of  E  will  be  real,  though  func- 
tions of  the  frequency  of  the  radiation  traversing  the  medium. 
In  vector  notation  P  =  \I/»E  (18) 

where  *  is  the  dyadic          +  e       +  e 


116      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

If  u  denotes  the  energy  per  unit  volume  of  the  medium, 
du  =  E^P 


Now  the  law  of  conservation   of  energy  requires  that  this 
expression  shall  be  an  exact  differential.    Therefore 

V".*' 

and  the  dyadic  x|/  is  self-conjugate.    Hence  by  a  proper  choice 
of  axes  ty  may  be  put  in  the  form 


Now  D  =  E  +  P 


Put  *=icji  +  Kv 

where  /c  =  1 


Then  D  =  *-E,  (19) 

showing  that  the  specific  inductive  capacity  is  a  dyadic  instead 
of  a  scalar  factor  as  in  the  case  of  an  isotropic  medium.  For 
"D  and  E  are  not  in  general  in  the  same  direction  in  a  medium 
which  is  not  isotropic.  Eliminating  B  and  D  from  equations  (1), 
(2),  (3),  (4),  and  (19),  it  is  found  that 

V-  VE  -  V  V-E  -  i  §-E  -  0.  (20) 

Confining  attention  to  plane  waves,  the  electric  intensity  and 
displacement  are  given  by  the  real  parts  of 


and  D= 
respectively. 

Therefore  dE=df  (^t»S)  E  -  dtiwE. 

But  dE  = 


ELECTROMAGNETIC  WAVES  117 


Comparing,  V  =  z 

and  —  =  —  i<*. 

ct 

Hence  (20)  becomes 

Sfi-E  +  SS-E  -  S-SE  =  0.  (21) 

Multiplying  by  S«,  it  is  seen  that 
S-D=0, 

or  the  vector  D  is  at  right  angles  to  the  direction  of  propa- 
gation of  the  wave.  Moreover,  equation  (21)  shows  that  D,  E, 
and  S  are  in  the  same  plane. 

The  magnetic  intensity  may  be  found  from  (2).    This  equation 


showing  that  L  is  perpendicular  to  both  S  and  E,  and  hence 
to  D.  The  vectors  D  and  L  lie  in  the  wave  front,  although  E 
makes  an  angle  with  this  plane.  The  flux  of  energy  is  given  by 

s  =  cE  x  L, 

which  is  a  vector  at  right  angles  to  E  in  the  plane  of  D,  E,  and  S. 
Fig.  18  shows  the  relative  directions  of  the  vectors  under 
discussion.  Since  the  flux  of  energy  is  not  along  the  wave 
normal,  limited  wave  fronts  will  side-  _ 

step  as  they  advance,  as  indicated  in  the 
figure.  A  line  drawn  in  the  direction 
of  the  flux  of  energy  is  known  as  a  ray, 
and  the  velocity  of  propagation  of  energy 
along  this  line  as  the  ray  velocity.  _ZI 


•D 


Consider  an  infinite  number  of  plane 
waves  passing  through  the  origin  at  the 
time  0  in  all  directions,  the  vector  D 

r  IG.  lo  JL* 

having  all  directions  in  the  wave  front. 

The  envelope  of  these  plane  waves  one  second  later  is  known 
as  the  Fresnel  wave  surface.  To  find  the  equation  of  this  sur- 
face, it  is  necessary  to  obtain  from  (21)  a  relation  involving 
the  wave  slowness  as  the  only  unknown  quantity ;  that  is,  an 


118   AN  INTRODUCTION  TO  ELECTRODYNAMICS 

equation  between  S,  S0,  and  §  which  is  true  for  all  possible 
directions  of  E.    Equation  (21)  may  be  written  in  the  form 

(S02*  +  SS  -  S-SI)  -E0  =  0,  (22) 

where  I  is  the  idemfactor 

ii  +  jj  +  kk. 

The  dyadic  in  the  parentheses  in  (22)  causes  the  vector  E0 
to  vanish.  Hence  either  its  antecedents  or  consequents  must  be 
coplanar.  This  dyadic  may  be  written  in  the  expanded  form 

-  S*  - 


-  -S1-  S,1)  J 


As  the  consequents  are  not  coplanar,  the  antecedents  must 
be.    Therefore  the  scalar  triple  product  of  the  latter  must  vanish. 

Hence  -  s 


2  s;s;s,*  =  o, 

or,  reducing, 

^,f  g.' 


If  ?,  m,  n  are  the  direction  cosines  of  the  wave  normal,  this 

equation  becomes 

72  wa  2 

*        _l_  _i_  =  Q  f  23^ 

T7-2          ^2     »     TZ2          12     '      T7-2          ^2  '  V         x 


where  F"  is  the  wave  velocity,  and 


ELECTROMAGNETIC  WAVES  119 

The   equation  of   a  plane  wave  advancing   in  the  direction 

specified  by  Z,  m,  n  is     7  ,r  .0/IN 

Ix  +  my  +  nz  =  V  (24) 

one  second  after  leaving  the  origin,  where  Z,  m,  n  satisfy  the 
condition  721      2  ,     ai  (9,^ 

The  Fresnel  wave  surface  is  the  envelope  of  the  family  of 
planes  obtained  by  varying  Z,  m,  n  in  equation  (24),  subject  to 
the  conditions  specified  in  (23)  and  (25).  This  surface  is  most 
easily  found  by  differentiating  (23),  (24),  and  (25),  and  elimi- 
nating Z,  m,  w,  and  V  by  means  of  the  original  equations.  The 
equations  obtained  by  differentiating  are 

xdl  +  ydm  +  zdn  =  dV,  (26) 

Idl  +  mdm  +  ndn  =  0,  (27) 

Idl          mdm         ndn     _  T,yjy  XOQ\ 

V2-a2      V2-b*      V2-<?  ~ 


Let  —  jt>  and  —  q  be  factors  by  which  (27)  and  (28)  respectively 
may  be  multiplied  so  as  to  eliminate  dl  and  dm  when  the  three 
equations  are  added  together.  As  the  other  differentials  are 
independent  of  one  another,  the  coefficient  of  each  differential 
in  the  sum  must  vanish,  giving 


(30) 


l=qkr.  (31) 

Multiplying  equations  (30)  by  Z,  m,  n  respectively,  adding, 
and  making  use  of  the  original  relations  (23),  (24),  (25),  it  is 

found  that  ,Qo\ 

p  =  V .  .  v."*J 


120      AN  INTRODUCTION  TO  ELECTRODYNAMICS 
Squaring  and  adding  the  two  sides  of  equations  (30), 


the  product  term  disappearing  on  account  of  (23).    Combining 
with  (31)  and  (32)  9  =  (r2  -  F2)  F.  (33) 

Substituting  in  (30)  the  values  of  p  and  q  just  found, 

V*fi*-V*)l      VI (r2-  a2) 
- * ^-  -  — ^ L  - 


or 


VI          x-Vl 


Vm 


Vn 


-Vm 


z  -  Vn 


(a) 


Multiplying  these  equations  by  ar, «/,  z  respectively  and  adding,  the 
required  equation        ~2  2  a 

pd^  +  p^  +  pri^=l  C34) 

of  the  Fresnel  wave  surface  is  obtained. 

In  discussing  this  surface,  it  will  be  considered  that 


The   trace   of   the   surface   on   the   YZ  coordinate   plane   is 
expressed  by 


which  is  a  circle  inside  an  ellipse,  as  in  Fig.  19  (a).    On  the  ZX 
plane  the  trace  is 


ELECTROMAGNETIC  WAVES 


121 


which  is  a  circle  cutting  an  ellipse,  as  in  Fig.  19(5).    Finally, 
the  trace  on  the  XY  plane  is  given  by 


which  is  an  ellipse  inside  a  circle,  as  in  Fig.  19(<?). 

One  octant  of  the  surface  is  represented  in  Fig.  20  (a),  and 
a  section  through  the  point  P  is  shown  in  20(6).  As  D  lies  in 
the  plane  of  the  wave  normal  and  the  ray,  this  vector  must  be 


FIG.  20 


tangent  to  each  of  the  elliptical  traces,  as  indicated  by  arrows 
in  the  combined  figure.  Consequently  it  must  be  perpendicular 
to  the  planes  of  the  circular  traces. 

The  primary  optic  axes  of  a  crystal  are  defined  as  those  direc- 
tions in  which  the  wave  velocity  is  independent  of  the  state  of 
polarization,  that  is,  the  direction  of  D  in  the  wave  front.  Hence 
the  perpendicular  OQ  to  the  tangent  QT  [Fig.  20  («)]  is  one 
of  the  primary  axes.  The  other  is  also  in  the  ZX  plane,  making 
an  equal  angle  with  OX  on  the  other  side  of  the  X  axis.  A 
crystal  which  has  two  optic  axes  is  known  as  biaxial.  Obviously, 
there  can  be  no  more  than  two  such  axes.  Since  the  constants 
which  determine  the  intercepts  of  the  Fresnel  surface  depend 
upon  the  three  principal  specific  inductive  capacities,  which  are 
themselves  functions  of  the  wave  length,  the  directions  of  the 
optic  axes  of  a  biaxial  crystal  vary  with  the  wave  length. 


122      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

The  secondary  optic  axes  are  defined  as  those  directions  in 
which  the  ray  velocity  is  independent  of  the  state  of  polari- 
zation. One  secondary  axis  has  the  direction  OP,  the  other 
making  an  equal  angle  with  OX  on  the  other  side  of  the  X  axis. 

A  uniaxial  crystal  is  one  in  which  two  of  the  quantities  a,  5,  c 
are  equal.  If  b  and  c  are  equal,  the  crystal  is  said  to  be  positive 
or  prolate.  The  Fresnel  wave  surface  is  shown  in  Fig.  21  (a). 


FIG.  21 


There  is  only  one  axis,  and  there  is  no  longer  any  distinction 
between  primary  and  secondary  axes.  Moreover,  the  direction 
of  this  axis  is  independent  of  the  wave  length. 

If  a  and  b  are  equal,  the  crystal  is  said  to  be  negative  or 
oblate.  The  Fresnel  wave  surface  for  such  a  crystal  is  shown  in 
Fig.  21  (6). 

45.  Reflection  and  refraction.  Consider  a  train  of  plane  waves 
incident  at  an  angle  </>x  (Fig.  22)  on  a  plane  surface  separating 
two  transparent  isotropic  media.  The  incident  light  will  be 
partly  reflected  and  partly  transmitted.  Let  Al  be  the  amplitude 
of  the  electric  vector  in  the  incident  radiation,  A[  that  in  the 
reflected,  and  Az  that  in  the  transmitted  radiation.  Then  the 
coefficient  of  reflection  R  is  defined  by 


and  the  coefficient  of  transmission  T  by 


ELECTROMAGNETIC  WAVES 


123 


Let  the  subscripts  1  and  2  refer  respectively  to  the  media 
above  and  below  the  plane  0  Y.  In  the  case  of  the  upper  medium, 
letters  without  primes  will  refer  to  the  incident  light,  and  letters 
with  primes  to  the  reflected  light.  Attention  will  be  confined 
to  electromagnetic  radiation  of  wave  length  long  compared  to 
the  distances  between  adjacent  molecules  of  either  medium. 
Hence  equations  (1)  to  (4)  inclusive  specify  the  field.  Moreover, 
if  the  media  under  consideration  are  transparent,  the  damping 
term  in  expression  (10)  for  the  specific 
inductive  capacity  is  negligible,  and 
this  quantity  is  real. 

Choose  axes  as  indicated  in  the 
figure,  the  Z  axis  extending  upward 
from  the  plane  of  the  paper.  Consider 
a  short  pill-box  shaped  surface,  with 
bases  parallel  to  the  YZ  plane  and  axis 
bisected  by  this  plane.  Integrating 
(1)  over  the  volume  enclosed  by  this 
surface,  and  transforming  the  volume 
integral  into  a  surface  integral  by 
means  of  Gauss'  theorem,  it  is  found 
that  the  components  of  D  normal  to  the  surface  of  separation 
are  the  same  on  the  two  sides  of  this  surface.  A  similar  relation 
between  the  normal  components  of  B  follows  from  (3). 

Consider  a  rectangle  of  which  one  pair  of  sides  is  very  much 
longer  than  the  other,  so  situated  that  the  short  sides  are  perpen- 
dicularly bisected  by  the  YZ  plane.  Integrating  (2)  over  the 
surface  bounded  by  this  rectangle,  and  transforming  the  left- 
hand  side  of  the  equation  into  a  line  integral  by  Stokes'  theo- 
rem, it  is  found  that  the  components  of  E  parallel  to  the  surface 
of  separation  are  the  same  on  the  two  sides  of  this  surface, 
provided  B  is  not  infinite  at  the  surface.  A  similar  relation 
between  the  parallel  components  of  L  follows  from  (4). 

Suppose  the  electric  vector  in  the  incident  wave  to  be  perpen- 
dicular to  the  plane  of  incidence ;  that  is,  the  light  is  polarized 
in  the  plane  of  incidence.  Then  the  x  and  y  components  of  the 


124      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

electric  intensity  are  zero  for  each  wave,  and  the  z  components 
are  given  by  the  real  parts  of 

E    —  kvi  eita * Si k cos <t>i  +  : "8in *t)-*} 

J£  f  —  fc^j  /gtaj{.S|(— x  COB  </>!  +  ?/ sin  <ft  ,•)  —  <} 
E2    =  k,4  ^  {  32  (*  cos  <f>2  +  ,/  sin  <J>2)  -  t }  m 
a 

Hence,  remembering  that  n  =  — 


and  that  the  permeability  is  unity  for  the  frequencies  under 
consideration,  it  follows  from  (2)  that 

LX  =  i^1^  sin  (f>l  —  jEJnl  cos  c^, 
Lf  =  iE'n  sin  <f>  -\-jEfn  cos  <f> , 
L2  =  iE0n2  sin  (£o  —  jEji^  cos  <£2, 

showing  that  L1  =  E1nv  L[  =  E[nv  and  L.2  =  E2n2. 

Therefore  the  relations  between  E  and  L  on  the  two  sides  of 
the  surface  of  separation  lead  to  the  three  equations 

V    _L   7?f         V  /Q<X\ 

-TV    ~r  ±2j    ==  J2j  ^  COO) 

1  1  2  V        / 

(Ev  +  ^)  nl  sin  <^>1  =  JE1^  sin  <£>2,  (36) 

(^  —  E'^)  n^  cos  ^>1  =  E2n.2  cos  (^>2.  (3  7) 

From  (35)  and  (36)  it  follows  that 

— 7~  ==  —  ~  "77  '  (^^/ 

sin  <£2      Wj      /S^ 
the  familiar  relation  attributed  to  Snell.    Dividing  (37)  by  (36), 

El  4-  EI      sin  <^>0  cos  ^ 

Now  the  exponentials  in  E^  and  E[  are  the  same  when  x  is 
zero.  Hence  the  electric  intensity  may  be  replaced  by  its  ampli- 
tude in  this  equation.  Thus 

1  —  R .       sin  6,  cos  cf> 


sn      cos 


ELECTROMAGNETIC  WAVES  125 

determines  the  coefficient  of  reflection  R^  for  the  case  where  the 
electric  vector  is  perpendicular  to  the  plane  of  incidence.    Solving, 

Sln  CQS  <       ~  Shl  C°S      >  Sln      <        ~ 


sin  <£2  cos  c/>x  -f-  sn  ^  cos  <2          sn  2 

To  find  the  coefficient  of  transmission  T±,  eliminate  E[  between 
(35)  and  (39).    Since  it  follows  from  (38)  that  the  exponentials 
in  E^  and  EZ  are  the  same  when  x  is  zero,  the  electric  intensity 
may  be  replaced  by  its  amplitude,  as  before.    Hence 
2  -  T±  _  sin  (^  cos  <ft2 
T±         sin  <f>2  cos  ^  ' 

2.sin<^  2cos<£ 

whence  T.  =  -r—  —  •  (41) 

sm^+c^,) 

In  the  case  where  the  electric  vector  is  in  the  plane  of  incidence, 
Ei  =  {-  Ui  sin  <t>1  +  j^j  cos 
Er=    -  U   sin  <>  -  ^   cos 


and  it  follows  from  (2)  that 


L2  =  k^2. 

In  this  case  the  relations  between  E  and  L  on  the  two  sides 
of  the  surface  of  separation  lead  to  the  three  equations 

'(^+  E'J  n*  sin  ^  =  Epl  sin  ^,  (42) 

(^  -  E()  cos  <k  =  ^2  cos  <^2,  (43) 

(^  +  ^)n1  =  ^2,  (44) 

since  K  =  n2. 

These  equations  lead  to  Snell's  law  and  the  following  ex- 
pressions for  the  coefficient  of  reflection  7?N  and  coefficient  of 
transmission  T,,, 

E  =  sin  ^  cos  ^  ~  s'm  ^  cos  ^  =  tan  (^i  ~  ^)  7      (45) 
sin  <^x  cos  (f>l  +  sin  <^>2  cos  <£2      tan  ((^j  -h  <^>2)  ' 

008^  46, 

' 


126      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Examination  of  the  four  coefficients  of  reflection  and  trans- 
mission shows  that  JBM  is  the  only  one  which  can  vanish.  The 
polarizing  angle  <&l  is  defined  as  the  angle  of  incidence  for  which 
this  coefficient  becomes  zero.  Therefore 

tan  (<l>1  -h  <l>2)  =  oo  , 
and  ^  and  3>2  are  complementary.    Consequently 

<l>  =  tan"1/  —  V 


Consider  unpolarized  light  striking  a  surface  at  the  polarizing 
angle.  The  energy  associated  with  the  component  of  the  electric 
vector  in  the  plane  of  incidence  will  be  entirely  transmitted. 
Consequently  the  reflected  light  will  consist  altogether  of  radia- 
tion in  which  the  electric  vector  is  at  right  angles  to  the  plane 
of  incidence.  Although  polarization  by  reflection  should  be 
complete  at  the  polarizing  angle,  experiment  does  not  show  it 
to  be  so.  This  is  largely  due  to  imperfect  surface  conditions. 

If  n^  is  greater  than  %2,  the  incident  radiation  may  be  totally 
reflected.  For  convenience  the  discussion  will  be  restricted  to 

the  case  for  which 

nl  =  n  >  1, 


Then,  for  total  reflection, 


and  sin  <f>2  =  n  sin  (f>1 

is  greater  than  unity,  and 


cos  </>2  =  i  V     sin2</>1  —  1 

is  imaginary.    Reference  to   (40)   and   (45)   shows  that  both 
coefficients  of  reflection  will  be  of  the  form 


Therefore 


ELECTROMAGNETIC  WAVES  127 

showing  that  total  reflection  has  produced  an  alteration  in  phase 
without  change  in  amplitude. 

Consider  the  transmitted  ray  in  the  case  of  total  reflection. 


and  as  cos<£2  is  imaginary, 


where  iy  =  S2  cos  </>2. 

This  is  a  wave  travelling  along  the  surface,  with  an  amplitude 
which  falls  off  exponentially  with  x.  As  the  wave  does  not 
pass  across  the  surface,  no  energy  is  taken  from  the  incident 
radiation. 

46.  Rotation  of  the  plane  of  polarization.  Consider  a  beam  of 
monochromatic  plane  polarized  light  travelling  through  a  trans- 
parent medium  in  the  direction  of  an  impressed  magnetic 
field  H.  Choose  axes  so  that  the  X  axis  is  parallel  to  H.  Fol- 
lowing the  method  developed  in  section  38,  it  is  found  that 
equation  (9),  section  37,  leads  to 


E= 


where  the  term  involving  the  damping  constant  I  in  the  denom- 
inator of  the  outstanding  factor  of  the  right-hand  side  has  been 
omitted  as  the  medium  is  transparent. 
Solving  for  R, 


e2 

< 
1 

where 


128      AN  INTRODUCTION  TO  ELECTRODYNAMICS 
Therefore,  dropping  the  strokes, 


Dz=KEz+i-eAHEy, 

G 

NneA 
where  K  =  1  H  --  -  -  , 


c 
NneA 


=  fc-l. 


- 

c 

Hence,  in  vector  notation 

B  =  A:E-i-€^ExH.  (47) 

C 

Since  the  electric  intensity  is  at  right  angles  to  the  direction 
of  propagation,  V-E  =  0 

and  elimination  of  B  from  the  field  equations  (1),  (2),  (3),  (4) 

gives  i  .. 

V-VE  =  ^D, 

c2 

the  relation  between  D  arid  E  being  specified  by  (47). 
Now  E  =  E 

Q2T? 

Therefore     V  •  VE  =  —  - 

dsr 


(48) 


fix 
and  =     -         E0  +  ieAE,  X  H 

Equating    real    and    imaginary   terms    in    these    two    equal 
expressions,         ^2p 

^  =  a>2(,S2-/c^)E0,  (49) 

ex 


ELECTROMAGNETIC  WAVES  129 

The  second  of  these  equations  shows  that  if  e  is  positive  the 
plane  of  polarization  rotates  in  the  counter-clockwise  sense  when 
viewed  from  the  source  of  light.  Such  rotation  is  called  positive. 
Reversing  the  direction  of  the  applied  magnetic  field  reverses 
the  sense  of  the  rotation.  Hence  if  a  beam  of  plane  polarized 
light  is  passed  through  a  transparent  body  along  the  lines  of 
force  of  an  applied  magnetic  field,  and  then  reflected  and  re- 
turned over  its  original  path,  the  rotation  of  the  plane  of 
polarization  is  not  annulled,  but  doubled.  This  magnetic  rota- 
tion was  discovered  experimentally  by  Faraday  in  1845. 

If  the  angle  rotated  through  is  denoted  by  a, 


da  = 


/SIN 


where  v  is  the  index  of  refraction.  Hence  the  rotation  varies 
with  the  strength  of  the  magnetic  field  and  the  length  of  the 
path.  In  the  neighborhood  of  an  absorption  band,  co  approaches 
&,  and  the  rotation  becomes  very  large.  If  the  vibrating  part 
of  the  atom  is  positively  charged,  the  rotation  will  be  positive 
when  the  direction  of  propagation  is  the  same  as  that  of  the 
magnetic  lines  of  force,  while  if  the  vibrating  part  of  the  atom 
is  negatively  charged,  the  rotation  will  be  negative.  Obviously, 
only  the  component  of  the  field  in  the  direction  of  propagation 
is  effective  in  producing  rotation. 

To  find  the  wave  slowness,  eliminate  E0  from  (49)  and  (50) 
and  solve  for  S.    Thus 


or,  approximately, 

(52) 


130   AN  INTRODUCTION  TO  ELECTRODYNAMICS 

47.  Metallic  reflection.  As  a  metal  contains  both  free  and 
bound  electrons,  the  equations  of  the  electromagnetic  field  take 
the  form 


=  -C  C 


where 


If  E 

then  V  =  i 


(53) 

V-B  =  0, 

(55) 

1     . 

(54)         ' 

c 

E),      (56) 

D  =  /cE, 

and  integration  with  respect  to  the  time  is  equivalent  to  multi- 
plication by  i/co.    Therefore  the  field  equations  become 

o,  V-B=0, 

1.  VxL  = 

xE=  --  B, 


G 


These  are  identical  with  equations  (1)  to  (4),  section  43,  for 
a  non-conducting  medium,  provided  the  specific  inductive  capacity 
is  replaced  by  the  complex  quantity 


o> 


Therefore,  remembering  that  the  permeability  is  unity  for 
light  frequencies,  the  wave  slowness  is  given  by 


(57) 


Hence,  if 
then 


and  E  =  E0e-"s"'Vw(s-r-<>.  (58) 


ELECTROMAGNETIC  WAVES 


131 


Let  the  YZ  plane  (Fig.  22,  p.  123)  be  the  surface  of  separation 
between  a  region  free  from  matter  above  this  plane  and  a 
metallic  medium  below.  Consider  a  train  of  plane  waves  inci- 
dent on  the  metallic  surface  at  an  angle  (^  Put 

A  =  ASHK 


C2  =  S  cos  <£2. 

Then  equations  (40)  and  (45),  section  45,  show  that 
7?        A%      r1  n 

-Ll,,  ./I O.  v_y_ 


(59) 


N 


A  and  Ct  being  real,  and  C72  complex.  This  ratio  may  be  put 
in  the  form  ^ 

I?1  =  />«""  (60) 

where  p  and  A  are  real. 

Suppose  the  incident  light  to  be  polarized  in  a  plane  making 
an  angle  of  45°  with  the  plane  of  incidence.  Such  radiation 
may  be  considered  to 
consist  of  two  trains  of 
waves,  one  polarized  in 
the  plane  of  incidence  <$, 

and  the  other  at  right 
angles  to  this  plane, 
which  have  the  same 
amplitude  and  are  in 
phase  with  each  other. 
After  reflection  these 
two  trains  of  waves  may 
have  different  ampli- 
tudes, and  their  phase 
relation  may  have  been 

changed  in  such  a  way  as  to  produce  elliptically  polarized  light. 
The  ratio  of  amplitudes  after  reflection  is  given  by  p  in  equa- 
tion (60),  and  the  difference  in  phase  by  A.  These  quantities 
may  be  conveniently  represented  by  means  of  a  graph  (Fig.  23) 


FIG.  23 


132      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

in  which  real  quantities  are  plotted  horizontally  and  imaginaries 
vertically.    Starting  from  the  origin  0,  lay  off  the  real  component 

OM=  icS* 
of  $2,  and  the  imaginary  component 

*-/      ^vo 


From  the  same  origin  lay  off  the  real  quantities 

OQ  =A\ 

QR  =  CI 

determined  by  the  angle  of  incidence.    Then 


and  Qp  =  c& 

is  laid  off  by  bisecting  the  angle  RQN  and  making 


Connecting  0  with  P  and  P\ 


OP  =  A 

±    -4 

Hence  the  ratio  of  amplitudes  after  reflection  is  given  by 


\OP'\  _1N 

p  =  l ',  (61) 

\OP\ 
and  the  difference  in  phase  between  the  two  components  by 

A  =  ZP'OP.  (62) 

48.  Zeeman  effect.  Consider  an  electron  which  may  vibrate 
under  the  influence  of  a  force  of  restitution  proportional  to  the 
electron's  displacement  from  its  position  of  equilibrium  inside 
the  atom.  The  equation  of  motion  of  such  an  electron  has  the 
components  ^2 


~ 


ELECTROMAGNETIC  WAVES  133 

showing  that  its  natural  vibration  has  a  frequency  defined  by 

T 


In  the  presence  of  a  magnetic  field  H  parallel  to  the  Z  axis 
the  electron  under  consideration  is  subject  to  an  additional  force 

-vxH, 

c 

so  that  the  components  of  the  equation  of  motion  become 

dzx          7        eHzdy 

m—=-kx-\ *-£, 

dt  c    dt 

d2y          ,         elf,  dx 

m  -~  =  —  ky -T- » 

d&  c    dt 

d*z 

m-—)  —  —Kz. 
dt2 

Solving  these  equations,  it  is  found  that 

x  =  A^os  (a>£  +  Sj)     and     y  =  —  Al  sin  (w^  +  ^),  (63) 

or        x  =  ^42cos(G)2«  +  82)     and     y  =  A2  sin  (t»2f  -f  8a),  (64) 

z  =  As  cos  (tot  +  S),  (65) 

e&> 
where  wr -1  If,  =  or, 

7?ZC 

/> 

or,  approximately,  ^  =  01  +  - —  Jfz ;  (66) 

„      e<»9  _  o 

and  a>22  +  — 2  JF/  =  w , 

we 

-.  =  —£;*,  (67) 

Equations  (63)  and  (64)  represent  rotation  in  circles  in  the 
XY  plane  in  the  negative  and  positive  senses  respectively  rela- 
tive to  the  Z  axis.  The  effect  of  the  magnetic  field  is  merely 
to  change  the  central  force  from 

kr 

to  kr^ej3irg. 


134      AN  INTRODUCTION  TO  ELECTRODYNAMICS 

Consider  a  body  which  emits  light  in  consequence  of  the 
vibrations  of  electrons  which  are  held  in  the  atoms  by  simple 
harmonic  forces  of  the  type  under  discussion.  Suppose  a  mag- 
netic field  to  be  applied  in  the  direction  of  the  Z  axis,  and  let 
the  source  of  light  be  viewed  along  the  Xaxis.  Vibrations  in 
the  X  direction  will  emit  no  radiation  in  the  direction  from 
which  the  light  is  being  observed.  Vibrations  in  the  Indirection 
will  give  rise  to  light  polarized  with  the  electric  vector  parallel 
to  the  Y  axis  of  frequencies  w^  and  o>2,  while  vibrations  in  the 
Z  direction  will  produce  light  polarized  with  the  electric  vector 
parallel  to  the  Z  axis  of  frequency  w.  Therefore  when  a  source 
of  light  in  a  magnetic  field  is  viewed  in  a  direction  at  right 
angles  to  the  lines  of  force,  each  spectral  line  will  be  resolved 
into  three  components.  The  central  undisplaced  component  will 
be  polarized  with  the  electric  vector  parallel  to  the  field,  and 
the  two  displaced  components  with  the  electric  vector  at  right 
angles  to  the  field. 

If  the  source  of  light  is  viewed  along  the  Z  axis,  no  radiation 
will  reach  the  observer  due  to  vibrations  parallel  to  this  axis. 
Vibrations  perpendicular  to  the  Z  axis  will  give  rise  to  circularly 
polarized  light  of  frequencies  o>1  and  o>2.  Consequently,  when  a 
source  of  light  in  a  magnetic  field  is  viewed  along  the  lines  of 
force,  each  line  will  be  resolved  into  two  components,  circularly 
polarized  in  opposite  senses  and  equally  displaced  on  either  side 
of  the  original  line.  There  will  be  no  undisplaced  component. 
The  sense  of  the  circular  polarization  of  the  two  displaced  com- 
ponents depends  upon  the  sign  of  the  vibrating  electrons,  which 
are  thus  shown  to  be  negative.  The  ratio  of  charge  to  mass 
of  the  negative  electron  may  be  calculated  from  the  displace- 
ments observed.  This  method  was  one  of  the  earliest  employed 
to  obtain  the  numerical  value  of  this  important  constant. 

While  the  results  obtained  from  theory  are  entirely  confirmed 
by  experiment  in  many  cases,  a  large  number  of  lines  are  split 
up  into  more  than  three  components  by  a  magnetic  field.  It  is 
believed  that  these  are  compound  lines,  which  the  optical 
apparatus  employed  is  not  powerful  enough  to  resolve. 

PRINTED  IN  THE   UNITED  STATES  OF  AMERICA 


ANNOUNCEMENTS 


PHYSICS  AND  CHEMISTRY 

PHYSICS 

Cavanagh,  Westcott,  and  Twining :  Physics  Laboratory  Manual 

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Higgins  :  Lessons  in  Physics 

Higgins  :  Simple  Experiments  in  Physics 

Hill :  Essentials  of  Physics 

Ingersoll  and  Zobel :  Mathematical  Theory  of  Heat  Conduction 

Jeans :  Theoretical  Mechanics 

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Miller :  Laboratory  Physics 

Millikan  :  Mechanics,  Molecular  Physics,  and  Heat 

Millikan  and  Gale:   First  Course  in  Physics  (Rev.  Ed.) 

Millikan,  Gale,  and  Bishop :  First  Course  in  Laboratory  Physics 

Millikan  and  Gale  :  Practical  Physics 

Millikan  and  Mills :  Electricity,  Sound,  and  Light 

Mills  :  Introduction  to  Thermodynamics 

Packard :  Everyday  Physics 

Snyder  and  Palmer :  One  Thousand  Problems  in  Physics 

Wentworth  and  Hill :  Textbook  in  Physics  (Rev.  Ed.) 

Wentworth  and  Hill:  Laboratory  Exercises  (Rev.  Ed.) 

CHEMISTRY 

Allyn  :  Elementary  Applied  Chemistry 

Dennis  and  Whittelsey:  Qualitative  Analysis  (Rev.  Ed.) 

Evans  :  Quantitative  Chemical  Analysis 

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McPherson  and  Henderson :  An  Elementary  Study  of  Chemistry 

(Second  Rev.  Ed.) 
McPherson  and  Henderson :  Course  in  General  Chemistry 

Laboratory  Manual  for  General  Chemistry 
McPherson  and  Henderson :  First  Course  in  Chemistry 
McPherson  and  Henderson  :   Laboratory  Exercises 
Moore  :  Logarithmic  Reduction  Tables 
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Olsen  :  Pure  Foods:  their  Adulteration,  Nutritive  Value,  and  Cost 
Sneed  :  Qualitative  Chemical  Analysis 
Test  and  McLaughlin :  Notes  on  Qualitative  Analysis 
Unger :  Review  Questions  and  Problems  in  Chemistry 
Williams :  Chemical  Exercises 
Williams  :  Essentials  of  Chemistry 


146 

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COLLEGE   PHYSICS 


MECHANICS,  MOLECULAR  PHYSICS,  AND  HEAT 

By  ROBERT  ANDREWS  MILLIKAN,  Professor  of  Physics  in  The  University 
of  Chicago.  8vo,  cloth,  242  pages,  illustrated. 

ELECTRICITY,  SOUND,  AND  LIGHT 

By  ROBERT  ANDREWS  MILLIKAN  and  JOHN  MILLS,  formerly  Professor 
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389  pages,  illustrated. 

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AN  ESSENTIALLY  practical  treatment  of  a  subject  of  vital  in- 
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